On Asymptotic Approximations of First Integrals for a Class of Nonlinear Oscillators

Transcription

1 On Asymptotic Approximations of First Integrals for a Class of Nonlinear Oscillators Stevanus Budi Waluya

2 For my son my wife my big family

3 On Asymptotic Approximations of First Integrals for a Class of Nonlinear Oscillators Proefschrift ter verkrijging van de graad van doctor aan de Technische Universiteit Delft op gezag van de Rector Magnificus prof. dr. ir. J. T. Fokkema voorzitter van het College voor Promoties in het openbaar te verdedigen op dinsdag 3 september 3 te 5.3 uur door Stevanus Budi Waluya Master of Science in Mathematics Institut Teknologi Bandung Indonesië geboren te Magelang Indonesië.

4 Dit proefschrift is goedgekeurd door de promotor: Prof. dr. ir. A.J. Hermans Toegevoegd promotor: Dr. ir. W.T. van Horssen Samenstelling promotiecommissie: Rector magnificus Prof. dr. ir. A.J. Hermans Dr. ir. W.T. van Horssen Prof. dr. ir. P.G. Bakker Prof. dr. ir. J. Blaauwendraad Prof. dr. A. Doelman Prof. dr. J. Molenaar Prof. dr. F. Verhulst voorzitter Technische Universiteit Delft promotor Technische Universiteit Delft toegevoegd promotor Technische Universiteit Delft Technische Universiteit Delft Universiteit van Amsterdam Universiteit Twente Rijks Universiteit Utrecht Waluya Stevanus Budi On Asymptotic Approximations of First Integrals for a Class of Nonlinear Oscillators / Thesis Delft University of Technology. With summary in Dutch. ISBN Copyright c 3 by S.B. Waluya All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means electronic or mechanical including photocopying recording or by any information storage and retrieval system without written permission from the copyright holder. Printed by [OPTIMA] Grafische Communicatie Rotterdam

5 Contents Introduction Asymptotic Approximations of First Integrals for a Nonlinear Oscillator 5. Introduction Integrating factors and an asymptotic theory Approximations of First Integrals The case b = O(ɛ) and c = ɛ The case b = O() and c = ɛ Approximations for time-periodic solutions Time-periodic solutions when b = O(ɛ) and c = ɛ Time-periodic solutions when b = O() and c = ɛ Conclusions and remarks On Approximations of First Integrals for Strongly Nonlinear Oscillators 5 3. Introduction Integrating vectors and an asymptotic theory Approximations of First Integrals Example of a Strongly Nonlinear Oscillator The case µ = and λ arbitrary The case µ > and λ arbitrary Time-periodic solutions and a bifurcation analysis The case µ = and λ arbitrary The case µ > and λ arbitrary Conclusions and remarks This chapter is a revised version of [4] Asymptotic Approximations of First Integrals for a Nonlinear Oscillator Nonlinear Analysis TMA 5(8): This chapter is a revised version of [45] On Approximations of First Integrals for Strongly Nonlinear Oscillators Nonlinear Dynamics 3 : i

6 ii CONTENTS 4 On Approximations of First Integrals for a System of Weakly Nonlinear Coupled Harmonic Oscillators Introduction Integrating vectors and an asymptotic theory Approximations of First Integrals The :3 internal resonance case The 3: internal resonance case Approximations for time-periodic solutions and analysis of bifurcations The :3 internal resonance case The 3: internal resonance case Analysis of Bifurcations Conclusions and remarks Appendix A - An oscillator with two degrees of freedom in a uniform windfield Appendix B - The :3 internal Resonance Appendix C - The 3: internal resonance case On Approximations of First Integrals for a Strongly Nonlinear Forced Oscillator Introduction Approximations of First Integrals Approximations of First Integrals for a Nonlinear Forced Oscillator The case λ = O(ɛ) The case λ = O( ɛ) The case λ = O() Time-periodic solutions and a bifurcation analysis The case λ = O(ɛ) The case λ = O( ɛ) The case λ = O() Conclusions and remarks Appendix Bibliography 3 Summary 7 Samenvatting 9 Acknowledgment Curriculum Vitae 3 This chapter is a revised version of [43] On Approximations of First Integrals for a System of Weakly Nonlinear Coupled Harmonic Oscillators Nonlinear Dynamics 3: This chapter is a revised version of [44] On Approximations of First Integrals for a Strongly Nonlinear Forced Oscillator to be published in Nonlinear Dynamics 3.

7 List of Figures. The oscillator as considered in this chapter Plot of Q(k ; b) as function of k for several values of b Sketch of the trajectories of system (.3.5) in the phase plane for several values of b Plot of the period T of the stable and unstable periodic solutions as functions of b Phase portrait of the unperturbed equation (3.4.) with ɛ = µ = and λ > Phase portrait of the unperturbed equation (3.4.8) with ɛ = µ = and λ < Plot of Q as function of E for µ = and for different values of λ Sketch of the appearance and disappearance of limit cycles for µ = and for decreasing values of the parameter λ Phase portrait of the unperturbed equation (3.4.7) with ɛ = for several values of ξ Plot of P as function of ξ for ξ < Plot of Q(E; ξ) as function of E for several values of ξ < Sketch of the appearance and disappearance of limit cycles for ξ = and for decreasing value of η Plot of R as function of ξ Plot of Q(E; ξ) as function of E for ξ = Plot of Q(E; ξ) as function of E for ξ = Plot of Q(E; ξ) as function of E for ξ = Sketch of the appearance and disappearance of limit cycles for ξ = 9 and for decreasing values of η Sketch of the appearance and disappearance of limit cycles for ξ = 9 and for decreasing values of η Plot of Q(E; ξ) as function of E for ξ = Sketch of the appearance and disappearance of limit cycles for 4 ξ < and for decreasing values of η Plot of Q(E; ξ) as function of E for ξ > Plot of amplitude A as function of ᾱ s iii

8 iv LIST OF FIGURES 4. Plot of the stable periodic solution for the :3 internal resonance case Plot of amplitudes A and A as function of ᾱ s for δ = Plots of amplitudes as functions of ᾱ s and δ Plots of the stable periodic solution for the 3: internal resonance case The aeroelastic oscillator as viewed from above Windvelocities and aerodynamic forces acting on the cross section of the cylinder with ridge The bifurcation diagram in the (β γ)-plane for the weakly nonlinear forced oscillator equation (5.3.) The bifurcation diagram in the (γ E )-plane for the weakly nonlinear forced oscillator equation (5.3.) with β = Phase Portraits in the (r ψ)-plane for the weakly nonlinear forced oscillator equation (5.3.) with β = and for several values of γ Poincaré-map results for the weakly nonlinear forced oscillator equation (5.3.) in the (X Ẋ)-plane for several values of γ with β = and ɛ = and with sample-times t equal to π 4π 6π 8π The bifurcation diagram in the (β γ)-plane for the nonlinear map (5.4.8) The bifurcation diagram in the (γ E )-plane for the nonlinear map (5.4.8) with β = Phase Portraits in the (r θ)-plane for the nonlinear forced oscillator equation (5.3.) with β = and for several values of γ Poincaré-map results for the nonlinear forced oscillator equation (5.3.) in the (X Ẋ)-plane for β = and for several values of γ and ɛ = 5 and with sample-times t = π + 4πn where n N The period T of the unperturbed equation (5.3.8) (that is (5.3.8) with ɛ = ) as function of the energy E = Ẋ + X + 4 X The curve in the (β γ)-plane for which the strongly nonlinear forced equation (5.3.8) has π-periodic solutions of order Poincaré-map results for the nonlinear forced oscillator equation (5.3.8) in the (X Ẋ)-plane for several values of γ with β = and ɛ = and with sample-times t = πn with n Z + for the figures (a) (c) (e) (f) and (h) and t = πn with n Z + for the figures (b) (d) (g) and (i)

9 List of Tables 3. Type of equilibrium points of the unperturbed equation (3.4.7) with ɛ = in the (Z Z ) phase plane v

10 Chapter Introduction In the period Euler developed the fundamental concept of how to make a single first-order ordinary differential equation (ODE) exact by means of integrating factors. It should be remarked that before 734 integrating factors were already known and used by for instance Leibniz and Bernoulli. Euler however showed that all integrating factors for a single first order ODE have to satisfy a first order partial differential equation. Also n th -order linear ODEs with constant coefficients were solved by Euler by means of an integrating factor. Lagrange extended Euler s method later to n th -order linear ODEs with non-constant coefficients. When the existence of solutions for a system of n first order ODEs has been established on a time-scale it is well-known that this system of n first order ODEs has n and can not have more than n functionally independent first integrals on this time-scale. A complete proof for this statement can be found in Forsyth [[3] p ]. Similar remarks on the existence of n functionally independent first integrals can also be found in for instance [ 4 8 9]. The fundamental concept to obtain these n first integrals by means of integrating vectors was recently presented in [3 33]. Based on this fundamental concept a perturbation method has been developed by Van Horssen in [ ] to study classes of weakly nonlinear regularly or singularly perturbed ODEs. When approximations of integrating vectors have been obtained an approximation of a first integral can be given. Also an error-estimate for this approximation of a first integral can be given on a time-scale. The main goal of the study as presented in this thesis is to investigate how the perturbation method based on integrating vectors can be extended to strongly nonlinear ODEs and to what classes of strongly nonlinear ODEs the method is applicable. The ODEs that will be studied in this thesis are all related to simple models which describe the galloping oscillations of overhead power transmission lines (on which for instance ice has accreted) in a windfield. Such lines may have cross sectional shapes which are aerodynamically unstable. As is well-known galloping of conductor lines is an almost purely vertical large amplitude low frequency oscillation of these lines. The mathematical models that describe these flow-induced vibrations of conductor lines in a uniform wind-field lead to weakly nonlinear or strongly nonlinear oscillator

11 Introduction equations of the type Ẍ + du(x) dx = ɛf(x Ẋ t) (..) where U(X) is the potential energy of the unperturbed (that is ɛ = ) nonlinear oscillator and where X = X(t) Ẋ = dx ɛ is a small parameter satisfying < ɛ dt and where f is a sufficiently smooth function. The functions X U and f can be vector-functions. To find the exact solution for (..) is usually impossible. For that reason perturbation methods are used to approximate the exact solution analytically. For weakly nonlinear equations a lot of perturbation methods (such as the multiple time-scales method the averaging method the harmonic balance method Melnikov s method the Poincaré-return map technique and many others) can be used to approximate the solution. For strongly nonlinear problems however the number of available perturbation methods becomes rather limited. In this thesis the recently developed perturbation method based on integrating vectors will be used to study the strongly nonlinear equation (..). In chapter the following oscillator equation is studied ) Ẍ + X + bx = c ( 3Ẋ Ẋ (..) where b and c are constants. The constant b is related to the nonlinear behavior of the springs and the constant c is related to flow force coefficients as has been explained by Van der Beek in [3]. Two cases will be considered: (i) b = O(ɛ) and c = ɛ and (ii) b = O() and c = ɛ. Asymptotic approximations of first integrals will be constructed and it will be shown how approximations of the exact solution of (..) can be obtained. Moreover it will be shown how the existence and the stability of time-periodic solutions of (..) can be obtained from the approximations of the first integrals. The first case also has been studied in [3] by using first order normal form techniques and the second case has also been studied in [] by using the Melnikov/Poincaré return-map technique. All results presented in chapter are in agreement with the results as obtained in [ 3]. In chapter 3 the following generalized Rayleigh oscillator equation is studied ) Ẍ + 9X + µx + λx 3 = ɛ (Ẋ Ẋ 3 (..3) where µ and λ are constant. In this chapter it will be shown how approximations of first integrals can be constructed for the generalized Rayleigh oscillator equation (..3). Not only approximations of the first integrals will be given in this chapter but it will also be shown how in a rather efficient way the existence and the stability of the time-periodic solutions can be obtained from these approximations. The bifurcation(s) of limit cycles will be studied in detail and a complete set of topological different phase portraits will be presented. It will also be shown in this chapter that at least five limit cycles can occur for the strongly nonlinear oscillator equation (..3).

12 3 In chapter 4 the following system of ODEs will be studied which describes the flow-induced vibrations of an oscillator with two degrees of freedom Ẍ + ( ω + ɛδ ) ] X = ɛ [ a Ẋ + a Ẏ + a Ẋ a ẊẎ + a Ẏ a 3 Ẏ 3 Ÿ + ( ω + ɛδ ) ] Y = ɛ [ b Ẋ + b Ẏ + b Ẋ b ẊẎ b Ẏ b 3 Ẏ 3 (..4) where a... a 3 b... b 3 ω ω δ and δ are constants. In [3] system (..4) has been studied for the ω : ω = : : and : internal resonance cases. First order normal form techniques and averaging techniques have been used in [3] to determine the existence and stability of nontrivial periodic solutions. In chapter 4 the recently developed perturbation method based on integrating vectors will be used to study system (..4) with a :3 and a 3: internal resonance. This chapter in fact completes the study of system (..4). As is well-known galloping is an almost purely vertical oscillation of conductor lines. In this chapter it will be shown that the system of oscillators will eventually oscillate in an almost purely vertical direction (that is perpendicular to the windflow). Finally in chapter 5 the following strongly nonlinear and nonautonomous oscillator equation is studied in detail ) Ẍ + X + λx 3 = ɛ (Ẋ β Ẋ 3 + γẋ cos(t) (..5) where λ > β > and γ are constants. In chapter 5 it will be shown how (..5) can be derived from a system of nonlinearly coupled oscillator equations describing the flow-induced vibrations of a cable in a windfield. Three cases will be studied in detail: (i) λ = O(ɛ) (ii) λ = O( ɛ) and (iii) λ = O(). In all cases asymptotic approximations of first integrals will be constructed. Using these approximations the existence and the stability of periodic solutions of (..5) will be investigated. Also the global behavior of the solutions of (..5) will be studied by using a phase-space analysis and the Poincaré-return map technique.

13 4 Introduction

14 Chapter Asymptotic Approximations of First Integrals for a Nonlinear Oscillator Abstract. In this chapter a generalized Rayleigh oscillator will be studied. It will be shown that the recently developed perturbation method based on integrating factors can be used to approximate first integrals and periodic solutions. The existence uniqueness and stability of time-periodic solutions are obtained by using the approximations for the first integrals.. Introduction The fundamental concept to reduce a first order ordinary differential equation to an exact one by means of integrating factors has been extended in [3] to systems of first order ordinary differential equations. In [33 34] a perturbation method based on these integrating factors has been presented for regularly perturbed ordinary differential equations (ODEs). When approximations of integrating factors have been obtained an approximation of a first integral can be given. Also an errorestimate for this approximation of a first integral can be given on a time-scale. It has also been shown in [33] how the existence and stability of time-periodic solutions for weakly nonlinear problems can be obtained from these approximations for the first integrals. In this chapter the recently developed perturbation method based on integrating factors is used to approximate first integrals and periodic solutions for a generalized nonlinear Rayleigh oscillator of the form ( ) Z + Z + bz = c Ż Ż (..) 3 This chapter is a revised version of [4] Asymptotic Approximations of First Integrals for a Nonlinear Oscillator Nonlinear Analysis TMA 5(8):

15 6 A Nonlinear Oscillator where Z = Z(t) and where b and c are constants. The dot represents differentiation with respect to t. In this chapter we consider two cases: (i) b = O(ɛ) and c = ɛ and (ii) b = O() and c = ɛ where ɛ is a small parameter satisfying < ɛ. In this chapter not only asymptotic approximations of first integrals are constructed but also asymptotic approximations of periodic solutions and their periods are determined. The presented results include existence uniqueness and stability properties of the periodic solutions. In [3] van der Beek uses (..) with b = O(ɛ) and c = ɛ as mathematical model to describe flow-induced vibrations of an oscillator with one degree of freedom in a uniform windfield. The oscillator and the frame are shown in Figure (.). The cylinder with ridge is rigidly attached to two shafts. These shafts ridge supports to the wall aircylinder bearings springs z y Figure.: The oscillator as considered in this chapter. two shafts can simultaneously move within two air-bearings in the z-direction. The springs provide the restoring forces in the z-direction. The constant windflow is in y-direction. The mathematical model that describes the flow-induced vibrations of the oscillator with one degree of freedom in a uniform wind-field can be given by ( ) Z + Z + ɛaz = ɛ Ż Ż (..) 3 where ɛ is a small parameter and where a is a constant of order one. In [3] it has been shown how ɛ and a depend on the physical quantities such as the windvelocity the aerodynamic drag and lift forces acting on the cylinder and so on. When the four springs are assumed to be nonlinear springs we can take (..) with b = O() and c = ɛ as a mathematical model to describe the flow-induced vibrations of the oscillator. Doelman and Verhulst [] Wiggins [46] and many others used the Melnikov or Poincaré-return map technique to study similar equations. In fact (..) has also been considered in [] using the Poincaré-return map technique. x

16 . Integrating factors and an asymptotic theory 7 However it has not been shown in [] that (..) has at most two limit cycles. Using the perturbation method based on integrating vectors and some numerical calculations we will give strong numerical evidence that equation (..) has at most two limit cycles. Moreover we will give in this chapter explicit approximations of the periods of the periodic solutions. In this chapter we show that straightforward expansions in ɛ can be used to construct asymptotic results on long time-scales. To obtain these results no classical perturbation techniques (such as averaging (see [8] [38]) or multiple (time) scales (see [] [34]) or Melnikov/Poincaré-return map techniques (see [] [5] [37] [46])) are used. This chapter is organized as follows. In section. of this chapter the perturbation methods based on integrating factors and an asymptotic theory will be given briefly. It will be shown in section.3 of this chapter how approximations of first integrals can be constructed. In section.4 it will be shown how the existence and stability of time-periodic solutions can be obtained. Finally in section.5 of this chapter some conclusions will be drawn and some remarks will be made.. Integrating factors and an asymptotic theory In this section we briefly outline the perturbation method based on integrating factors as given in [ ]. We consider the system of n first order ODEs (n ) dy dt = f(y t) (..) where f = (f f... f n ) T y = (y y... y n ) T and where the superscript T indicates the transposed. For each i i n y i = y i (t) and f i = f i (y y... y n t). We assume that f f... f n are sufficiently smooth such that a twice continuously differentiable first integral F (y y... y n t; c) = (..) exists where c is an arbitrary constant. Furthermore we assume that there exist continuously differentiable integrating factors µ µ... µ n with µ i = µ i (y y... y n t) i i =... n. To obtain the relationships between the first integral and the integrating factors we multiply each i-th ODE in (..) with the integrating factor µ i and we then add the so-obtained equations yielding µ dy dt = µ f(y t) (..3) where µ = (µ µ... µ n ) T. In fact µ can be considered as an integrating vector. This exact ODE should be the same equation as the equation obtained by differentiating (..) with respect to t. That is µ dy + µ dt dy µ dt n dyn (µ f) = and dt F y dy + F dt y dy F dt y n dyn + F = dt dt (..4)

17 8 A Nonlinear Oscillator are equivalent and we can write F = µ (..5) F = µ f t ( ) T where = y y... y n. Eliminating F from (..5) by differentiations we obtain µ i y j = µ j y i i < j n (..6) µ = (µ f). t All integrating vectors µ for the system of ODEs (..) have to satisfy the system of n(n + ) first order linear PDEs (..6). Now we consider the following system of n first order ODEs dy dt = f(y t; ɛ) (..7) where ɛ is a small parameter and where the function f has the form f(y t; ɛ) = f (y t) + ɛf (y t). (..8) An integrating vector µ = µ(y t; ɛ) for system (..7) has to satisfy (..6). Assume that µ can be expanded in a power series in ɛ that is µ(y t; ɛ) = µ (y t) + ɛµ (y t) ɛ m µ m (y t) (..9) We determine an integrating vector up to O(ɛ m ). An approximation F app of F in the first integral F = constant can be obtained from: F app = µ + ɛµ ɛ m µ m ) ] (..) F app = [(µ t + ɛµ ɛ m µ m f where the * indicates that terms of order ɛ m+ and higher have been neglected. Then we obtain F app (y t; ɛ) = F (y t) + ɛf (y t) ɛ m F m (y t). (..) It should be observed that an approximation up to O(ɛ m ) of an integrating vector µ has been used to obtain an exact ODE up to O(ɛ m+ ) that is df app dt = d dt (F + ɛf ɛ m F m ) = F app dy dt + F app t ) ] = [(µ + ɛ µ ɛ m µ m f = ɛ m+ R m+ (y t µ... µ m ; ɛ) (..)

18 .3 Approximations of First Integrals 9 where the ** indicates that only terms of order ɛ m+ and higher are included. Let initial values for problem (..7) be given for t =. Then for ɛ it follows from (..) that (when R m+ is bounded): or t F app (y(t) t; ɛ) F app (y() ; ɛ) = ɛ m+ R m+ (...)dt F app (y t; ɛ) = constant + O(ɛ m+ ) t T (..3) F app (y t; ɛ) = constant + O(ɛ m ) t L ɛ (..4) where T and L are ɛ-independent constants..3 Approximations of First Integrals In this section we will show how the perturbation method based on integrating factors can be applied to approximate first integrals for a generalized Rayleigh oscillator. In the first part of this section we will consider the linear perturbed case and in the second part the nonlinear perturbed case..3. The case b = O(ɛ) and c = ɛ Consider the mathematical model which describes the flow-induced vibrations of an oscillator with one degree of freedom in a uniform windfield Z + Z + ɛaz = ɛ ( ) Ż 3 Ż (.3.) where < ɛ and a >. To analyze equation (.3.) the equation is first written as a system of first order ODEs. Let X = Z X = Ż from (.3.) we obtain X = X X = X ɛax + ɛ ( X 3 ) X. (.3.) In polar coordinates r and θ (where X = r cos θ and X = r sin θ) system (.3.) becomes ( ) dr = ɛar cos θ sin θ + ɛ r sin θ r sin θ = f dt 3 (r θ) ( dθ = ɛar cos 3 θ + ɛ dt r sin θ 3 ) cos θ sin θ = f (r θ). (.3.3)

19 A Nonlinear Oscillator Multiplying the first equation in (.3.3) by µ and the second one by µ respectively it follows from (..6) that the integrating factors µ and µ have to satisfy µ = µ θ r µ = (µ t r f + µ f ) (.3.4) µ t = θ (µ f + µ f ). Expanding µ and µ in power series in ɛ that is µ (r θ t; ɛ) = µ (r θ t) + ɛµ (r θ t) + ɛ µ (r θ t) +... µ (r θ t; ɛ) = µ (r θ t) + ɛµ (r θ t) + ɛ µ (r θ t) +... (.3.5) substituting f f and the expansions for µ and µ into (.3.4) and by taking together terms of equal powers in ɛ we finally obtain µ = µ θ r O(ɛ µ ) : = µ (.3.6) t r µ = µ t θ O(ɛ ) : µ θ = µ r and O(ɛ n ) with n : µ n = µ n θ r µ n t = r µ t = r (µ ar cos θ sin θ + µ + µ ar cos 3 θ) µ t = θ (µ ar cos θ sin θ + µ + µ ar cos 3 θ) (µ n ar cos θ sin θ µ n ( r sin θ 3 ) r sin θ ) ) +µ n + µ n ar cos 3 θ µ n ( r sin θ cos θ sin θ 3 ( µ n = (µ t θ n ar cos θ sin θ µ n r sin θ 3 ) r sin θ ) ) +µ n + µ n ar cos 3 θ µ n ( r sin θ cos θ sin θ. 3 (.3.7) (.3.8) The O(ɛ )-problem (.3.6) can be solved yielding µ = h (r θ + t) and µ = h (r θ + t) with h = h. The functions h θ r and h are still arbitrary and will now be chosen as simple as possible. We choose h and h and so µ = µ =. (.3.9)

20 .3 Approximations of First Integrals The O(ɛ )-problem (.3.7) then becomes: µ = µ θ r µ = ar cos θ sin θ + µ t r µ = ar cos θ sin θ + ar cos 3 θ + µ. t θ The O(ɛ )-problem (.3.) can also readily be solved yielding µ = h (r θ + t) + ar 3 cos3 θ µ = h (r θ + t) ar sin θ + ar sin 3 θ (.3.) (.3.) with h = h. The functions h θ r and h are still arbitrary. We choose these functions as simple as possible: h h. And we obtain µ = ar 3 cos3 θ (.3.) µ = ar sin θ + ar sin 3 θ. The O(ɛ )-problem (.3.8) now becomes µ = µ θ r µ t = r µ t = θ ( 3 a r 3 cos 5 θ sin θ ( r sin θ 3 ) r sin θ + ( ar sin θ + ar sin 3 θ ) ar cos 3 θ + µ ) ( ( 3 a r 3 cos 5 θ sin θ r sin θ 3 ) r sin θ + ( ar sin θ + ar sin 3 θ ) ar cos 3 θ + µ ). (.3.3) The solution of (.3.3) is given by µ = h (r θ + t) + ( 3r) θ sin θ r sin θ 5 64 a r cos θ 3 r sin 4θ 3 a r cos 4θ 9 a r cos 6θ µ = h (r θ + t) r cos θ + 6 r3 cos θ a r 3 sin θ r3 cos 4θ a r 3 sin 4θ + 96 a r 3 sin 6θ (.3.4) with ( 3r) + h = h. The functions h 8 θ r and h are still arbitrary. We choose these functions as simple as possible h h r 8 r3 (.3.5)

21 A Nonlinear Oscillator and we obtain µ = ( θ sin θ r sin θ 5 64 a r cos θ 3 r sin 4θ 3 a r cos 4θ 9 a r cos 6θ µ = ( r 8 r3) r cos θ + 6 r3 cos θ a r 3 sin θ r3 cos 4θ a r 3 sin 4θ + 96 a r 3 sin 6θ. (.3.6) The O(ɛ n )-problems (.3.8) with n > can be solved in a similar way. An approximation F of a first integral F = constant can be obtained from (.3.9) (.3.) (.3.6) and (..) yielding F (r θ t; ɛ) = r + {( 3 ɛar cos 3 θ + ɛ r ) 8 r3 θ r sin θ 4 + r3 sin θ 5 9 a r 3 cos θ 96 r3 sin 4θ 96 a r 3 cos 4θ } 576 a r 3 cos 6θ. (.3.7) How well F approximates F in a first integral F = constant follows from (..)- (..4). In this case we have df dt = [(µ + ɛµ ɛ m µ m ) ] ) ] f [(µ + ɛµ ɛ m µ m f = [ f + ɛµ f + ɛ µ f + ɛµ f + ɛ µ f ] = ɛ 3 R 3 (r θ) (.3.8) where f f µ µ µ µ are given by (.3.3)(.3.) and (.3.6). From the existence and uniqueness theorems for ODEs we know that an initial-value problem for system (.3.) is well-posed on a time-scale of order. This implies that also ɛ an initial-value problem for system (.3.3) is well-posed on this time-scale. From (.3.3) it then follows on this time-scale that if r() is bounded then r(t) is bounded and θ(t) is bounded by a constant plus t. Since R 3 c + c t on a time scale of order where c ɛ c are constants it follows from (.3.8) that and so t F (r(t) θ(t) t; ɛ) = constant + ɛ 3 R 3 (r(s) θ(s) s; ɛ)ds (.3.9) F (r(t) θ(t) t; ɛ) = constant + O(ɛ 3 ) t T < F (r(t) θ(t) t; ɛ) = constant + O(ɛ) t L ɛ (.3.) where T and L are ɛ-independent constants. In a similar way we can construct a second (functionally independent) approximation F of a first integral by putting µ = µ = (.3.)

22 .3 Approximations of First Integrals 3 in (.3.6) instead of (.3.9) µ = a sin θ + a 3 sin3 θ µ = ar cos 3 θ (.3.) in (.3.7) instead of (.3.) and so on. After some elementary calculations we then find ( F (r θ t; ɛ) = θ + t + ɛ ar sin θ + a ) { 5 3 r sin3 θ + ɛ a r θ + 4 r cos θ 3 cos θ a r sin θ 96 r cos 4θ + 48 a r sin 4θ + } 88 a r sin 6θ. (.3.3).3. The case b = O() and c = ɛ We consider in this subsection ( Z + Z + bz = ɛ Ż 3 ) Ż (.3.4) where b > and < ɛ. Let Z = X Ż = X from (.3.4) we obtain X = X = g (X X ) X = X bx + ɛ ( X 3 ) X = g (X X ). (.3.5) Multiplying the first equation in (.3.5) by the integrating factor µ and the second one by µ it follows from (..6) that the integrating factors µ and µ have to satisfy µ X = µ X µ t = X (µ g + µ g ) (.3.6) µ t = X (µ g + µ g ). For a time-independent first integral the integrating factors µ and µ have to satisfy µ g + µ g = (.3.7) µ X = µ X where µ = µ (X X ; ɛ) and µ = µ (X X ; ɛ). We assume that the integrating factors µ and µ can be expanded in power series in ɛ that is µ = µ (X X ) + ɛµ (X X ) +... (.3.8) µ = µ (X X ) + ɛµ (X X ) +....

23 4 A Nonlinear Oscillator The expansions (.3.8) are substituted into (.3.7) and terms of equal power in ɛ are taken together yielding µ O(ɛ X = µ X ) : (.3.9) µ X + µ ( X bx ) = µ X = µ X O(ɛ ) : µ X + µ ( X bx ) + µ and the O(ɛ n )-terms with n are µ n X = µ n X ( X 3 ) X = (.3.3) µ n X + µ n ( X 3 ) X + µ n ( X bx ) =. (.3.3) From (.3.9) it follows that µ ( ) X + bx µ + X X X X ( ) X + bx µ =. (.3.3) Using the method of characteristics we easily find the general solution of the PDE (.3.3) ( µ = f X + X + ) 3 bx3 X (.3.33) where f is an arbitrary function. From (.3.9) µ then also easily follows. Now take µ and µ as simple as possible for instance µ = X (.3.34) µ = X + bx. Using (.3.34) the O(ɛ )-problem (.3.3) can then be rewritten as µ ( ) X + bx µ + X X X X ( ) X + bx µ + ( ) X =. (.3.35) The corresponding characteristic ODEs of this PDE (.3.35) are dx = ds dx = ds X (X + bx ) dµ = (X ds X + bx ) µ ( X ). It follows from (.3.36) that dx dx = X X + bx (.3.36) X + 3 bx3 + X = k (.3.37)

24 .3 Approximations of First Integrals 5 dµ dx = X ( ) X + bx µ ( ) X (.3.38) where k is a constant of integration. Using (.3.37) it follows from (.3.38) that the solution of the homogeneous equation (.3.38) is ( µ h = C k X ) 3 bx3 (.3.39) where C is constant. The general solution of the inhomogeneous equation (.3.38) then easily follows by applying the method of variation of parameter yielding X ( + k r 3 µ = br3) ( dr + k (k r 3 br3) k X ) 3 bx3 (.3.4) where k is a constant of integration. From (.3.3) we obtain µ = ( ) ( X + bx ) µ X X3 X 3 X ( + k r 3 = br3) ( ) ( dr + k X + bx ) (k r X 3 br3 ) X3. 3 (.3.4) In (.3.4) and (.3.4) k can be considered as an arbitrary function of k and so k = g ( X + 3 bx3 + ) X where g is an arbitrary function. We will take g since we are interested in approximations of first integrals that are as simple as possible. An approximation F of F in a time-independent first integral F (X X ; ɛ) = constant can be obtained from (.3.34) (.3.4) (.3.4) and (..5) and F = µ + ɛµ yielding F t = F (X X ; ɛ) = X + 3 bx3 + [ X + ɛ X 4 + ( k + X + ) 3 bx3 X ( + k r ) ( 3 br3 3 X (.3.4) X ( + k r 3 br3) (k dr r br3) 3 ( + k r br3) 3 (k dr r br3) 3 (k r 3 ) ) br3 ] dr. (.3.43)

25 6 A Nonlinear Oscillator How well F approximates F can be shown by differentiating F with respect to t that is df = ɛ ( ) X X X3 X + k r 3 br3 dr.(.3.44) dt 3 k r 3 br3 From the existence and uniqueness theorems for ODEs we know that initial value problems for system (.3.5) are well-posed on t T <. This implies that system (.3.44) is well-posed on t T. So we have F (X X ) = constant + O(ɛ ) t T (.3.45) where T is an ɛ-independent constant..4 Approximations for time-periodic solutions In section.3 we constructed asymptotic approximations of first integrals. In this section we will show how the existence the stability and the approximations of non-trivial time-periodic solutions can be determined from these asymptotic approximations of the first integrals..4. Time-periodic solutions when b = O(ɛ) and c = ɛ Using these approximations it is possible to study the existence and stability of timeperiodic solutions. Let T < be the period of a periodic solution and let c be a constant in the first integral F (r θ t; ɛ) = constant for which a periodic solution exists. Consider F = c for t = nt and t = (n )T with n N + then F (r(nt ) θ(t ) nt ; ɛ) = c (.4.) F (r ((n )T ) θ((n )T ) (n )T ; ɛ) = c. For the autonomous equation (.3.3) we may assume without loss of generality that θ() =. From (.3.3) it follows that r(nt ) = r ((n )T ) + O(ɛ) (.4.) θ(nt ) = θ ((n )T ) T + O(ɛ). Approximating F by F (given by (.3.7)) eliminating c from (.4.) and using (.4.) we obtain ( r(nt ) = r ((n )T ) + ɛ T r ((n )T ) ) 8 r3 ((n )T ) + O(ɛ 3 t). (.4.3) on a time scale of order ɛ. In fact (.4.3) defines a map Q : r Q(r) r n = Q(r n ) with r n = r(nt ). We define a new map P by neglecting the term of O(ɛ 3 t) in (.4.3). That is P : r P ( r) r n = P ( r n ) with r n = r(nt ). It will be shown that for r > :

26 .4 Approximations for time-periodic solutions 7 (i) If r r = O(ɛ) for ɛ then r n r n = O(ɛ) for n = O ( ɛ ) that is for n ɛ and ɛ r n and r n remain ɛ-close. (ii) The map P has a unique hyperbolic fixed point r = which is asymptotically stable. (iii) There exists an ɛ > such that for all < ɛ ɛ the map Q has a unique hyperbolic fixed point r = + O(ɛ) with the same stability property as the fixed point r = of the map P. P roof of (i): From r r = O(ɛ) for ɛ it follows that there exists a positive constant M such that r r = M ɛ. We have r n r n = P (r n ) P ( r n ) + O(ɛ 3 n) P (r n ) P ( r n ) + M ɛ 3 n L r n r n + M ɛ 3 n (.4.4) where M and L are positive constants with L = + ɛ M and M a positive constant. So we have r n r n ( + ɛ M ) r n r n + M ɛ 3 n... ɛ(m + ɛ n M )e ɛ nm (.4.5) and so for n = O( ɛ ) we conclude that r n r n = O(ɛ). P roof of (ii): The fixed points of the map P follow from r n = P ( r n ) for n or equivalent from r = r + ɛ T ( r 8 r3 ) r( 4 r ) =. For r > we have a unique fixed point r =. The fixed point of the map P is hyperbolic if the linearized map around this fixed point has no eigenvalues of unit modulus. Let DP be this linearized map then DP = ɛ T. Since < ɛ we have λ < and so the fixed point is hyperbolic and stable. P roof of (iii): For the proof of (iii) we refer to [33] for a similar proof. So far we conclude that there exists a unique asymptotically stable nontrivial T -periodic solution for system (.3.). We can approximate the form and the period T of the limit cycle of system (.3.) from F (r θ t; ɛ) = c where F is approximated by F. For the periodic solution we have r() = r(t ) and θ(t ) = θ() π. Without loss of generality we assume that θ() = and since the system (.3.) is autonomous we know that a time-independent first integral exists. So we have to solve the following system of two non-linear equations F (r() ; ɛ) = c (.4.6) F (r(t ) π; ɛ) = c

27 8 A Nonlinear Oscillator where r() = r(t ) T and c are three unknown constants. Now it should be observed that F given by (.3.7) approximates this time-independent first integral up to O(ɛ 3 ) for t T. Without loss of generality it can be assumed that the unknown constants r() = r(t ) and c can be approximated by r +ɛr +ɛ r +... and c + ɛc + ɛ c +... respectively. By substituting these approximations for r() = r(t ) and c into (.4.6) and by collecting terms of equal powers in ɛ we obtain the constants r r... c c... after some elementary calculations yielding r() = r(t ) = + ɛ ( 4 3 a) + ɛ ( 8 9 a) + O(ɛ 3 ) c = + ɛ ( a) + O(ɛ 3 ). (.4.7) From F (r θ t; ɛ) = c where F is approximated by the time-independent function F given by (.3.7) the radius r of the limit cycle can be approximated as function of θ yielding { r(θ) = + ɛ 4 } { 37 3 a cos3 θ + ɛ 36 a a cos3 θ sin θ sin θ a cos θ + sin 4θ + a cos 4θ + } 7 a cos 6θ + O(ɛ 3 ). (.4.8) By substituting (.4.8) into (.3.3) and by expanding θ(t) in θ (t)+ɛθ (t)+ɛ θ (t)+... and then by expanding the right hand side of (.3.3) in ɛ the functions θ (t) θ (t)... can be calculated yielding θ(t) = t + ɛ { 3 a sin t 6 } { a sin 3t + ɛ a t a sin t 4 a sin 4t 7 a sin 6t + cos t + } cos 4t + O(ɛ 3 ). (.4.9) 4 The period T of the limit cycle can be approximated from θ(t ) = π. Now let T = T + ɛt + ɛ T +... and substituting θ(t ) = π into (.4.9) we obtain T = π a (π)ɛ + O(ɛ 3 ). (.4.) We also can approximate the periodic solution of (.3.) that is Z(t) = r(t) cos(θ(t)) where r(t) and θ(t) are approximated by (.4.8) and (.4.9)..4. Time-periodic solutions when b = O() and c = ɛ Let T < be the period of a periodic solution and let c be the constant in the first integral F = constant for which a periodic solution exists. Consider F = c for t = and t = T. Approximating F by F (given by.3.43) and eliminating c by subtraction we obtain { X (T ) ɛ X () ( k r ) ( (k 3 br3 r 3 ) ) 3 br3 dr } = O(ɛ ) ɛi(k ; b) + O(ɛ ) = (.4.)

28 .4 Approximations for time-periodic solutions 9 where I(k ; b) = B A ( k r ) ( (k 3 br3 r 3 ) ) 3 br3 dr (.4.) with A = X () and B = X ( T ). To have a periodic solution for (.3.4) we have to find a constant k such that I(k ; b) is equal to zero. To find this constant k we rewrite I(k ; b) in ( I(k ; b) = I (k ; b) I ) (k ; b) (.4.3) I (k ; b) where I (k ; b) = B A I (k ; b) = 3 B A ( k r 3 br3) dr > ( k r 3 br3) 3 dr >. (.4.4) Now it should be observed that the unperturbed equation (.3.4) with ɛ = has two equilibrium points: a center point in (Z Ż) = ( ) and a saddle point in (Z Ż) = ( ). The center point trajectory and the saddle point loop are b represented by k = and k = respectively. It should be observed that if k 6b then A and B and B A I (k ; b) lim k I (k ; b) = lim 3 k B A And if k 6b ( k r 3 br3) 3 dr ( k r 3 br3) dr then A b and B b and =. (.4.5) lim k 6b I (k ; b) I (k ; b) = lim k 6b = 3 b b b b B 3 A B A ( k r 3 br3) 3 dr ( k r 3 br3) dr ( ) r + 3 ( b br + ) 3 dr 3 3 ( ) ( r + b br + ) dr 3 3 = 6 77b. (.4.6) Let Q(k ; b) = I (k ;b) I (k. Then it can be shown analytically that ;b) and dq (; b) = > (.4.7) dk ( ) dq dk 6b ; b =. (.4.8) For several values of b (see Figure.) we have calculated Q(k ; b) numerically. The integrals I (k ; b) and I (k ; b) have been determined by using an adaptive recursive Simpson rule and the boundary points (A and B) have been determined by using a Newton rule. The phase plane portraits are given for several values of b in Figure.3. Using numerical calculations (see also Figure.) we are able to find k -values such that I(k ; b) = or Q(k ; b) =. We obtain the following results:

29 A Nonlinear Oscillator (a) b=. (b) b= (c) b= (d) b=.9 Figure.: Plot of Q(k ; b) as function of k for several values of b. 6 (i) for b < 77 +O(ɛ ) there is only one nontrivial value of k such that I(k ; b) =. This implies that there exists only one periodic solution. 6 (ii) for + 77 O(ɛ 6 ) b + δ there are two nontrivial values for k 77 such that I(k ; b) =. This implies that there are two periodic solutions. It should be remarked that δ is very small; δ is approximately equal to (iii) for b > + δ there are no values of k 77 such that I(k ; b) = and so there are no periodic solutions. It should be noted that these bifurcation results are in accordance with those obtained in []. To determine the stability of the time-periodic solutions we consider the approximation F of a first integral F = X + 3 bx3 + X + ɛ { X ( (k r 3 ) )} 3 br3 ( k r ) 3 br3 dr

30 .4 Approximations for time-periodic solutions Z Z Z Z (a) b=. (b) b= Z Z Z Z (c) b= (d) b=.9 Figure.3: Sketch of the trajectories of system (.3.5) in the phase plane for several values of b. where Ĩ = Ĩ = 3 = X + 3 bx3 + X + ɛ { Ĩ X (t) X () X (t) X () (k r 3 br3 ) dr ( Ĩ Ĩ )} (.4.9) (k r 3 br3 ) 3 dr. (.4.) In the (X X )-phase plane we now define the positive X -axis to be a Poincaré section. At t = we start in (X () ) with X () >. After some time (t = t ) we return to the Poincaré section that is (X (t ) X (t )) = (X (t ) ) with X (t ) >. From (.4.) and (.4.9) we know that X (t) + 3 bx3 (t) + ( X (t) = k + ɛ (I I )) + O(ɛ ). (.4.) I

31 A Nonlinear Oscillator For a periodic solution Q = I I should be and (.4.) then becomes X (t) + 3 bx3 (t) + X (t) = k + O(ɛ ) (.4.) where k is assumed to be the constant in the approximation of a first integral for which a periodic solution exists. From the numerical calculations of Q = I I (see also Figure.) it follows that there are two possibilities: (i) I I < for k < k and I I > for k > k or (ii) I I > for k < k and I I < for k > k. In the first case (i) it follows from (.4.) that if ( )) ɛ (I I I ( then ɛ (I I > and so X (t ) > X (). And similarly if I I I I < for k < k then > for k > k )) I < and so X (t ) < X (). So in the first case (i) the periodic solution is stable. In case (ii) it can be shown similarly that the periodic solution is unstable. The period T of the periodic solution can be determined from or from dx dt T = ( = X = k X ) 3 bx3 (.4.3) B A dx (k X ) (.4.4) 3 bx3 where < A < B < and where A B satisfy b b X + 3 bx3 = k. It should be observed that if k then A and B so 6b b b B dx T = lim (k k 6b A X ) 3 bx3 = = b dx ( ) ( b X + b X ( [ 4 tanh 3 br + 3 ) ) ] b b =. (.4.5) Using (.4.4) we have determined T (b) numerically (see Figure.4)..5 Conclusions and remarks In this chapter it has been shown that the perturbation method based on integrating factors can be used efficiently to approximate first integrals for a generalized Rayleigh oscillator. The method can also be applied to other nonlinear oscillator equations

32 .5 Conclusions and remarks 3 (a) Stable (b) Unstable (d) Stable and Unstable Figure.4: Plot of the period T of the stable and unstable periodic solutions as functions of b. that are integrable when the small parameter is zero. In section. (and.3) of this chapter an asymptotic justification of the presented perturbation method has been given. For a generalized Rayleigh oscillator it has been shown how the existence and stability of time-periodic solutions can be deduced from the approximations of the first integrals. For the nonlinear Rayleigh oscillator it has been shown that 6 there exists one stable periodic solution if < b < + 77 O(ɛ ) that there are two 6 periodic solutions (one stable and one unstable) if + 77 O(ɛ 6 ) b + δ (δ is 77 approximately ) and that there are no periodic solutions if b > + δ. 77

33 4 A Nonlinear Oscillator

34 Chapter 3 On Approximations of First Integrals for Strongly Nonlinear Oscillators Abstract. In this chapter strongly nonlinear oscillator equations will be studied. It will be shown that the recently developed perturbation method based on integrating factors can be used to approximate first integrals. Not only approximations of first integrals will be given but it will also be shown how in a rather efficient way the existence and stability of time-periodic solutions can be obtained from these approximations. In particular ) the generalized Rayleigh oscillator equation Ẍ +9X + µx + λx 3 = ɛ (Ẋ Ẋ 3 will be studied in detail and it will be shown that at least five limit cycles can occur. 3. Introduction In [ ] a perturbation method based on integrating factors and vectors has been presented for regularly or singularly perturbed systems of ordinary differential equations (ODEs). When approximations of integrating vectors have been obtained an approximation of a first integral can be given. Also an errorestimate for this approximation of a first integral can be given on a time-scale. It has also been shown in [ ] how in a rather efficient way the existence and stability of time-periodic solutions can be obtained from these approximations for the first integrals. In this chapter it will be shown explicitly how the perturbation method can be applied to the following strongly nonlinear oscillator equation Ẍ + c X + c X + c 3 X 3 = ɛf(x Ẋ) (3..) This chapter is a revised version of [45] On Approximations of First Integrals for Strongly Nonlinear Oscillators Nonlinear Dynamics 3:

35 6 Strongly nonlinear oscillators where c c c 3 are parameters where < ɛ and where the dot represents differentiation with respect to t. Recently equation (3..) obtained a lot of attention. For example in [] Doelman and Verhulst studied (3..) with f(x Ẋ) = ( X ) Ẋ (a Van der Pol type of perturbation) by using a Melnikov/Poincaré return map technique. For c = and f(x Ẋ) = bẋ Yuste and Bejarano [49] applied a Krylov-Bogoliubov method. Coppola and Rand [] and Roy [9] used an averaging method which is based on elliptic functions to study (3..) with c =. Also for c = Chen and Cheung [9 ] used a Lindstedt-Poincaré method to study (3..) with f(x Ẋ) = (a bx ) Ẋ where a and b are constants. By using Melnikov functions and a Picard-Fuchs analysis Iliev and Perko [7] studied (3..) with c = c = ± c 3 = ± and f(x Ẋ) = aẋ + bx + cxẋ + dx Ẋ where a b c and d are parameters. For equations like (3..) Blows and Perko [4] wrote an interesting survey paper on Melnikov/Poincaré techniques. Margallo and Bejarano [4 5] and Lynch [3] studied (3..) with c = and f(x Ẋ) = Ẋ Ẋ3 and showed that at least one limit cycle can occur. Waluya and Van Horssen [4] studied (3..) with c 3 = and f(x Ẋ) = Ẋ Ẋ3 (a Rayleigh type of perturbation) by using the perturbation method based on integrating factors. It has been shown in [ 4] that for (3..) with c 3 = and f(x Ẋ) = Ẋ Ẋ3 two limit cycles can occur. The case c 3 c and f(x Ẋ) = Ẋ Ẋ3 has not yet been studied and will be studied in this chapter using the perturbation method based on integrating factors. It will turn out in this chapter that five limit cycles can occur. Equation (3..) with f(x Ẋ) = Ẋ Ẋ3 plays an important role in applications for instance in flow-induced vibrations of cables in a windfield. For details of this application we refer the readers to the papers of Van der Beek [3 3]. Using the perturbation method based on integrating vectors and some numerical calculations we will study (3..) with c 3 and c and f(x Ẋ) = Ẋ Ẋ3 in detail that is the existence and stability and the bifurcation of time-periodic solutions will be investigated in detail. In this chapter we restrict ourselves to autonomous differential equations. The presented perturbation method however can also be extended to nonautonomous equations. In relation to these nonautonomous equations we refer the readers for instance to the work of Roy [9] Brothers and Haberman [6] and Bosley [5] who used averaging and matching techniques to obtain insight in the solution structure for a class of non-autonomous equations. This chapter is organized as follows. In section 3. of this chapter the perturbation method based on integrating vectors and an asymptotic theory will be given briefly. It will be shown in section 3.3 of this chapter how approximations of first integrals can be constructed for the strongly nonlinear oscillator equation Ẍ + du(x) dx = ɛf(x Ẋ) (3..) where U(X) is the potential energy of the unperturbed (that is ɛ = ) nonlinear oscillator and where X = X(t) Ẋ = dx ɛ is a small parameter satisfying < ɛ dt and where f is a sufficiently smooth function. Approximations of first integrals for the oscillator equation (3..) with f(x Ẋ) = Ẋ Ẋ3 will be presented in section 3.4 of this chapter. Using these approximations it will be shown in section 3.5 how the existence and stability of time-periodic solutions for the oscillator equation (3..)

36 3. Integrating vectors and an asymptotic theory 7 with f(x Ẋ) = Ẋ Ẋ3 can be obtained. The bifurcation(s) of limit cycles will be studied in detail and a complete set of topological different phase portraits will be presented. Finally in section 3.6 of this chapter some conclusions will be drawn and some remarks will be made. 3. Integrating vectors and an asymptotic theory In this section we briefly outline the perturbation method based on integrating vectors as given in [ ]. We consider the following system of n first order ODEs dy dt = f(y t; ɛ) (3..) where ɛ is a small parameter and where the function f has the form f(y t; ɛ) = f (y t) + ɛf (y t). An integrating vector µ = µ(y t; ɛ) for system (3..) has to satisfy µ i y j = µ j y i i < j n µ t = (µ f). (3..) Assume that µ can be expanded in a power series in ɛ that is µ(y t; ɛ) = µ (y t) + ɛµ (y t) ɛ m µ m (y t) We determine an integrating vector up to O(ɛ m ). An approximation F app of F in the first integral F = constant can be obtained from: F app = µ + ɛµ ɛ m µ m ) ] F app = [(µ t + ɛµ ɛ m µ m f (3..3) where the * indicates that terms of order ɛ m+ and higher have been neglected. Then we obtain F app (y t; ɛ) = F (y t) + ɛf (y t) ɛ m F m (y t). It should be observed that an approximation up to O(ɛ m ) of an integrating vector µ has been used to obtain an exact ODE up to O(ɛ m+ ) that is df app dt = ) ] [(µ + ɛ µ ɛ m µ m f = ɛ m+ R m+ (y t µ... µ m ; ɛ) (3..4) where the ** indicates that only terms of order ɛ m+ and higher are included. How well F app approximates F (y t; ɛ) = constant can be determined from (3..4) that is error estimates can be given on time-scales depending on the boundedness properties of R m+.

37 8 Strongly nonlinear oscillators 3.3 Approximations of First Integrals In this section we will show how the perturbation method based on integrating vectors can be applied to approximate first integrals for a strongly nonlinear oscillator equation. We consider the class of non-linear oscillators described by the equation Ẍ + du(x) dx = ɛf(x Ẋ) (3.3.) where U(X) is a potential X = X(t) Ẋ = dx ɛ is a small parameter satisfying dt < ɛ and where f is assumed to be sufficiently smooth. We assume that the unperturbed (that is ɛ = ) solutions of (3.3.) form a family of periodic orbits. This family may cover the entire phase plane (X Ẋ) or a bounded region D of the phase plane. Each periodic orbit corresponds to a constant energy level E = + U(X). With each constant energy level E corresponds a phase angle ψ which Ẋ is defined to be ψ = X dr E U(r). (3.3.) From (3.3.)-(3.3.) a transformation (X Ẋ) (E ψ) can then be defined with Ė = ɛẋf = g (E ψ) [ ψ = + ɛ X dr (E U(r)) 3 ] Ẋf = g (E ψ). (3.3.3) Multiplying the first and the second equation in (3.3.3) with µ (E ψ t) and µ (E ψ t) respectively it follows from (3..) that the integrating factors µ (E ψ t) and µ (E ψ t) have to satisfy µ = µ ψ E µ t µ t = E (µ g + µ g ) = ψ (µ g + µ g ). Expanding µ and µ in formal power series in ɛ that is µ i (E ψ t; ɛ) = µ i (E ψ t) + ɛµ i (E ψ t) +... (3.3.4) for i = and substituting g g and the expansions for µ and µ into (3.3.4) and by taking together terms of equal powers in ɛ we finally obtain the following O(ɛ n )-problems: for n = µ ψ = µ E µ t = µ E (3.3.5) µ t = µ ψ

38 3.3 Approximations of First Integrals 9 and for n µ n = µ n ψ E µ n t µ n t = E (µ n g + µ n g + µ n ) = ψ (µ n g + µ n g + µ n ) (3.3.6) where ɛg = g ɛg = g. The O(ɛ )-problem (3.3.5) can readily be solved yielding µ = h (E ψ t) and µ = h (E ψ t) with h = h. The ψ E functions h and h are still arbitrary and will now be chosen as simple as possible. We choose h and h and so (see also [33 4]) µ = µ =. (3.3.7) Then it follows from the order ɛ-problem (3.3.6) that µ and µ have to satisfy µ + µ = (g t ψ E ) µ + µ t ψ = ψ (g ). By using the method of characteristics for first order PDEs we then obtain µ = h (E ψ t) t ( (g E ) ) d t µ = h (E ψ t) ( ) t (g ψ ) d t where h h are arbitrary functions which have to satisfy h ψ t ( ) ψ E (g ) d t = h E t ( ) E ψ (g ) (3.3.8) (3.3.9) d t. (3.3.) We choose h and h as simple as possible that is we take h = h =. We then obtain for µ and µ ( µ = t g E d t) ( µ = t g ψ d t). (3.3.) An approximation F of a first integral F = constant of system (3.3.3) can now be obtained from (3.3.7) (3.3.) and (3..3) yielding [ t ] F (E ψ t) = E ɛ g d t. (3.3.) How well F approximates a first integral F = constant follows from (3..4). In this case we have df dt = [( + ɛµ )g + ɛµ g ] = ɛµ g + ɛµ (g ) = ɛ R (E ψ t) (3.3.3)

39 3 Strongly nonlinear oscillators where g g and µ µ are given by (3.3.3) and (3.3.) respectively. From the existence and uniqueness theorems for ODEs we know that initial value problems for (3.3.) (with sufficiently smooth potential U(X) and nonlinearity f(x Ẋ)) are well-posed on a time-scale of order. This implies that also an initial-value problem ɛ for system (3.3.3) is well-posed on this time-scale. From (3.3.3) it then follows on this time-scale that if E() is bounded then E(t) is bounded and ψ(t) is bounded by a constant plus t. Since R c + c t on a time scale of order where c ɛ c are constants it follows from (3.3.3) that and so t F (E(t) ψ(t) t; ɛ) = constant + ɛ R (E(s) ψ(s) s; ɛ)ds F (E(t) ψ(t) t; ɛ) = constant + O(ɛ ) t T < F (E(t) ψ(t) t; ɛ) = constant + O(ɛ) t L ɛ (3.3.4) where T and L are ɛ-independent constants. It is well-known (see also section 3 in [43] for proofs and references) that a system of two first order ODEs has two and cannot have more than two functionally independent first integrals. Another (functionally independent) approximation of a first integral can be obtained by putting in (3.3.5) µ = µ = (3.3.5) instead of (3.3.7). The O(ɛ)-problem (3.3.6) can now be solved yielding µ = k (E ψ t) t ( (g E ) ) d t µ = k (E ψ t) ( ) t (g ψ ) d t (3.3.6) where the functions k and k are arbitrary functions which have to satisfy k ψ t ( ) ψ E (g ) d t = k E t ( ) E ψ (g ) d t. (3.3.7) We choose these functions as simple as possible that is k = and k =. Then we obtain ( µ = t g E d t) ( ) µ = t g ψ d t. (3.3.8) An approximation F of a first integral F = constant of system (3.3.3) can now be obtained from (3.3.5) (3.3.8) and (3..3) yielding [ t ] F (E ψ t) = (ψ t) ɛ g d t. (3.3.9)

40 3.4 Example of a Strongly Nonlinear Oscillator 3 How well F approximates a first integral F = constant follows from (3..4). In this case we have df dt = [ɛµ g + ( + ɛµ )g ] = ɛµ g + ɛµ (g ) = ɛ R (E ψ t) where g g and µ µ are given by (3.3.3) and (3.3.8) respectively. In the following section we will treat some examples to show how this perturbation method can be applied. 3.4 Example of a Strongly Nonlinear Oscillator In this section we will consider the following strongly nonlinear oscillator equation Ẍ + du(x) dx where du(x) dx = ɛf(x Ẋ) (3.4.) = 9X + µx + λx 3 with µ and λ parameters where the function f(x Ẋ) is a so-called Rayleigh perturbation that is f(x Ẋ) = Ẋ Ẋ3 and where ɛ is a small parameter with < ɛ. In [3] Van der Beek introduced (3.4.) with µ = O( ɛ) and λ = as a model equation to describe the vibrations of an oscillator in a uniform windfield. This model equation is related to the phenomenon of galloping of overhead power transmission lines on which ice has accreted. Using first order normal form techniques it has been shown in [3] that (3.4.) with µ = O( ɛ) and λ = has a unique (stable) periodic solution. Doelman and Verhulst [] and Waluya and Van Horssen [4] showed that a stable and an unstable periodic solution can occur simultaneously for (3.4.) with µ = O() and λ =. For (3.4.) with µ = and λ > Garcia-Margallo and Bejarano [5] showed that at least one limit cycle can occur. In this section we will construct approximations of first integrals for (3.4.) with µ and λ arbitrary. To give a complete analysis of (3.4.) we have to consider two main cases: (i) µ = and λ arbitrary and (ii) µ > and λ arbitrary. It should be observed that the case µ < and λ arbitrary is included in case (ii) (just replace X by X in (3.4.)). The constructed approximations of the first integrals will be used in section 3.5 to determine the number of periodic solutions for (3.4.) The case µ = and λ arbitrary To study (3.4.) with µ = in detail we have to consider three subcases: λ = λ > and λ <. These three cases will be studied in the following three subsections. The case µ = and λ = By putting X(t) = X(τ) with t = τ 3 Rayleigh oscillator equation is obtained in (3.4.) the following weakly nonlinear X + X = ɛg( X ) (3.4.)

41 3 Strongly nonlinear oscillators where ɛ = ɛ X = d X and where g( X ) = X 9( X ) 3. By introducing the 3 dτ transformation ( X X ) (E ψ) as defined by E = ( X ) + ( X) ψ = X dr E r ( ) = sin E X (3.4.3) (where E and ψ are the energy and the phase angle of the unperturbed oscillator (ɛ = ) respectively) we obtain the following system of ODEs E = ɛ X g = ξ (E ψ) = ɛξ (E ψ) [ ψ = + ɛ X dr (E r ) 3 ] X g = ξ (E ψ) = + ɛξ (E ψ). (3.4.4) From the calculations as presented in section 3.3 of this chapter it follows that two functionally independent approximations of first integrals for system (3.4.4) are given by τ τ F (E ψ τ) = E ɛ ξ d t = E ɛ (( X ) 9( X ) ) 4 d t and τ ( = E ɛ E cos(ψ) 36E cos(ψ) 4) d t (( = E ɛ E 7 ) E ψ 9E sin(ψ) 9 ) 8 E sin(4ψ) (3.4.5) τ F (E ψ τ) = (ψ τ) ɛ ξ d t = (ψ τ) + ɛ τ ( E sin(ψ) cos(ψ) 36E sin(ψ) cos(ψ) 3) d t E ( = (ψ τ) + ɛ 4 cos(ψ) + 4 E cos(ψ) + ) 6 E cos(4ψ). (3.4.6) How well F and F approximate a first integral F = constant follows from (3..4). In this case for j = we have df j = ɛµ ξ + ɛµ (ξ ) = ɛ R j (E ψ) (3.4.7) dτ where ξ and ξ are given by (3.4.4). It follows from (3.4.7) that for j = (see also (3.3.3)-(3.3.4)) and so τ F j (E(τ) ψ(τ) τ; ɛ) = constant + ɛ R j (E(s) ψ(s) s; ɛ)ds (3.4.8) F j (E(τ) ψ(τ) τ; ɛ) = constant + O( ɛ ) τ T < F j (E(τ) ψ(τ) τ; ɛ) = constant + O( ɛ) τ L ɛ (3.4.9) where T and L are ɛ-independent constants.

42 3.4 Example of a Strongly Nonlinear Oscillator 33 The case µ = and λ > By putting X(t) = X(τ) with t = τ 3 oscillator equation is obtained in (3.4.) the following nonlinear Rayleigh X + X + β X 3 = ɛg( X ) (3.4.) where ɛ = ɛ β = λ 3 9 X = d X and where g( X ) = X 9( X ) 3. By introducing the dτ transformation ( X X ) (E ψ) as defined by E = ( X ) + ( X) + β( X) 4 4 ψ = X (3.4.) dr E r βr4 (where E and ψ are the energy and the phase angle of the unperturbed oscillator (that is (3.4.) with ɛ = )) we obtain the following system of ODEs E = ɛ X g = ξ 3 (E ψ) = ɛξ 3 (E ψ) [ ψ = + ɛ X dr (E r βr4 ) 3 ] X g = ξ 4 (E ψ) = + ɛξ 4 (E ψ). (3.4.) The solution of the unperturbed equation (3.4.) is X = A cn(ϑ k) with ϑ = ω ψ where ψ = τ +constant k is a modulus given by k = βa and ω ω = +βa (see also [8 9 9]). The relationship between the energy E and the amplitude A is given by E = A + 4 βa4. The function cn(ϑ k) is a Jacobian elliptic function with argument ϑ and modulus k. From the calculations as presented in section 3.3 of this chapter it follows that two functionally independent approximations of first integrals for system (3.4.) are given by τ τ F 3 (E ψ τ) = E ɛ ξ 3 d t = E ɛ (( X ) 9( X ) ) 4 d t and = E ɛ [ τ (ω A sn(ϑ k) dn(ϑ k) ηω 4 A4 sn(ϑ k)4 dn(ϑ k) 4 ) dϑ F 4 (E ψ τ) = (ψ τ) ɛ τ ξ 4 d t = (ψ τ) + ɛ [ τ P (ϑ k) (ω A sn(ϑ k)dn(ϑ k) ω ] (3.4.3) ] ηωa 3 3 sn(ϑ k) 3 dn(ϑ k) 3) dϑ (3.4.4) ω where P (ϑ k) = A cn(ϑ k) A E ψsn(ϑ k)dn(ϑ k) ω + A E k sn(ϑ k) and dn(ϑ k) are elliptic functions and where A by A E = A + βa 3 ω E = βa ω (A + βa 3 ) ω E E cn(ϑ k) k E and k E k E = βa ( k ) kω (A + βa 3 ). in which are given

43 34 Strongly nonlinear oscillators How well F 3 and F 4 approximate a first integral F = constant follows from (3..4). In this case for j = 3 4 we have df j = ɛµ ξ 3 + ɛµ (ξ 4 ) = ɛ R j (E ψ) (3.4.5) dτ where ξ 3 and ξ 4 are given by (3.4.). It follows from (3.4.5) that for j = 3 4 (see also (3.3.3)-(3.3.4)) and so τ F j (E(τ) ψ(τ) τ; ɛ) = constant + ɛ R j (E(s) ψ(s) s; ɛ)ds (3.4.6) F j (E(τ) ψ(τ) τ; ɛ) = constant + O( ɛ ) τ T < F j (E(τ) ψ(τ) τ; ɛ) = constant + O( ɛ) τ L ɛ (3.4.7) where T and L are ɛ-independent constants. The case µ = and λ < By putting X(t) = X(τ) with t = τ 3 equation is obtained in (3.4.) the following nonlinear oscillator X + X γ X 3 = ɛg( X ) (3.4.8) where γ = λ > ɛ = ɛ X = d X and where g( X ) = X 9( X ) 3. By 9 3 dτ introducing the transformation (X X ) (E ψ) as defined by E = ( X ) + ( X) γ( X) 4 4 ψ = X (3.4.9) dr E r + γr4 (where E and ψ are the energy and the phase angle of the unperturbed oscillator (that is (3.4.) with ɛ = )) we obtain the following system of ODEs E = ɛ X g = ξ 5 (E ψ) = ɛξ 5 (E ψ) [ ψ = + ɛ X dr (E r + γr4 ) 3 ] X g = ξ 6 (E ψ) = + ɛξ 6 (E ψ). (3.4.) The solution of the unperturbed equation (3.4.8) is given by X = A sn(ϑ k) with ϑ = ω ψ where ψ = t + constant k = γa ω ω = γa and E = A 4 γa4. From the calculations as presented in section 3.3 of this chapter it follows that two functionally independent approximations of first integrals for system (3.4.) are given by τ τ F 5 (E ψ τ) = E ɛ ξ 5 d t = E ɛ (( X ) 9( X ) ) 4 d t = E ɛ [ τ (ω A cn(ϑ k) dn(ϑ k) ω 4 A 4 cn(ϑ k) 4 dn(ϑ k) 4 ) dϑ ω ] (3.4.)

44 3.4 Example of a Strongly Nonlinear Oscillator 35 and F 6 (E ψ τ) = (ψ τ) ɛ τ ξ 6 d t = (ψ τ) + ɛ [ τ P (ϑ k) (ω A cn(ϑ k)dn(ϑ k) ηω 3 A3 cn(ϑ k)3 dn(ϑ k) 3) ] dϑ (3.4.) ω where P (ϑ k) = A sn(ϑ k) + A E ψcn(ϑ k)dn(ϑ k) ω + A E k sn(ϑ k) k E A E = A γa 3 ω E = γa ω (A γa 3 ) k E = γa ( + k ) kω (A γa 3 ). in which How well F 5 and F 6 approximate a first integral F = constant follows from (3..4). In this case for j = 5 6 we have df j dτ = ɛµ ξ 5 + ɛµ (ξ 6 ) = ɛ R j (E ψ) (3.4.3) where ξ 5 and ξ 6 are given by (3.4.). It follows from (3.4.3) that for j = 5 6 (see also (3.3.3)-(3.3.4)) and so τ F j (E(τ) ψ(τ) τ; ɛ) = constant + ɛ R j (E(s) ψ(s) s; ɛ)ds (3.4.4) F j (E(τ) ψ(τ) τ; ɛ) = constant + O( ɛ ) τ T < F j (E(τ) ψ(τ) τ; ɛ) = constant + O( ɛ) τ L ɛ (3.4.5) where T and L are ɛ-independent constants The case µ > and λ arbitrary By putting X(t) = 9 µ Z(τ) t = τ 3. (3.4.6) in (3.4.) the following nonlinear oscillator equation is obtained Z + Z + Z + ξz 3 = ɛg(z ) (3.4.7) where ξ = 9λ µ ɛ = 3 ɛ Z = dz dτ and g(z ) = Z η(z ) 3 with η = 93 µ. By introducing the transformation (Z Z ) (E ψ) as defined by E = Z + Z + 3 Z3 + 4 ξz4 ψ = Z (3.4.8) dr E r 3 r3 ξr4

45 36 Strongly nonlinear oscillators (where E and ψ are the energy and the phase angle of the unperturbed oscillator (that is (3.4.) with ɛ = )) we obtain the following system of ODEs E = ɛz g = ζ (E ψ) = ɛζ [ ψ = + ɛ Z dr (E r 3 r3 ξr4 ) 3 ] Z g = ζ (E ψ) = + ɛζ. (3.4.9) From the calculations as presented in section 3.3 of this chapter it follows (see also section 3.4.) that two functionally independent approximations of first integrals for system (3.4.9) are given by F 7 (E ψ) = E ɛ τ ζ d t τ = E ɛ (G [] ηg4 [] )d t (3.4.3) and F 8 (E ψ) = (ψ τ) ɛ τ ζ d t τ = (ψ τ) ɛ F [](G [] ηg 3 [])d t (3.4.3) where F [] = Z E and G [] = Z are elliptic functions which are defined by (3.4.8). How well F 7 and F 8 approximate a first integral F = constant follows from (3..4). For j = 7 8 we have df j dτ = ɛµ ζ + ɛµ (ζ ) = ɛ R j (E ψ) (3.4.3) where ζ and ζ are given by (3.4.9). It follows from (3.4.3) that for j = 7 8 (see also (3.3.3)-(3.3.4)) and so τ F j (E(τ) ψ(τ) τ; ɛ) = constant + ɛ R j (E(s) ψ(s) s; ɛ)ds (3.4.33) F j (E(τ) ψ(τ) τ; ɛ) = constant + O( ɛ ) τ T < F j (E(τ) ψ(τ) τ; ɛ) = constant + O( ɛ) τ L ɛ (3.4.34) where T and L are ɛ-independent constants.

46 3.5 Time-periodic solutions and a bifurcation analysis Time-periodic solutions and a bifurcation analysis In the previous section it has been shown how asymptotic approximations of first integrals can be obtained. In this section we will show how the existence of non-trivial time-periodic solutions can be determined from the asymptotic approximations of the first integrals. We will also present phase portraits and a bifurcation analysis. In section 3.5. we will show that (3.4.) with µ = can have (at least) two limit cycles and in section 3.5. we will give strong numerical evidence that (3.4.) with µ can have (at least) five limit cycles The case µ = and λ arbitrary To determine the non-trivial time-periodic solutions from the asymptotic approximations of the first integrals we have to consider (as in section 3.4.) three subcases: λ = λ > and λ <. These three cases will be studied in the following three subsections. The case µ = and λ = Let T < be the period of a periodic solution and let c be a constant in the first integral F (E ψ τ ; ɛ) = constant for which a periodic solution exists. Consider F = c for τ = and τ = T. Approximating F by F (as given by (3.4.5)) eliminating c by subtraction we then obtain (using the fact that E() = E(T ) for a periodic solution) ( T ɛ (( X ) 9( X ) ) ( ) X(T ) ( ) 4 d t = O( ɛ ) ɛ X 9( X ) ) 3 d X = O( ɛ ). X() (3.5.) Because of the symmetry of the unperturbed orbits in the phase plane (3.5.) can be rewritten as where ɛi(e) = O( ɛ ) (3.5.) I(E) = 4 A ( X 9( X ) 3 ) d X (3.5.3) with A = X( T ) = E. To have a periodic solution for (3.4.) we have to find an energy E such that I(E) is equal to zero (see also [ ]). To find this energy E we rewrite I(E) in ( I(E) = 4I (E) 9 I ) (E) (3.5.4) I (E)

47 38 Strongly nonlinear oscillators where I (E) = A I (E) = A (E X ) d X = Eπ (E X ) 3 d X = 3E π 4. (3.5.5) It now easily follows from (3.5.4) and (3.5.5) that I(E) = for E = or E =. 7 Putting Q = I I (where I and I are as defined in (3.5.5)) it easily follows from (3.5.5) that Q = 3 E. Since Q is strictly monotonically increasing in E we can conclude that there exists a unique nontrivial stable periodic solution for (3.4.). The standard arguments leading to this conclusion can for instance be found in [[4] section 4. ] or in [4 7]. The case µ = and λ > Let T < be the period of a periodic solution and let c be a constant in the first integral F (E ψ τ ; ɛ) = constant for which a periodic solution exists. Consider F = c for τ = and τ = T. Approximating F by F 3 (as given by (3.4.3)) eliminating c by subtraction we then obtain (using the fact that E() = E(T ) for a periodic solution) ( T ɛ (( X ) 9( X ) ) ( ) X(T ) ( ) 4 d t = O( ɛ ) ɛ X 9( X ) ) 3 d X = O( ɛ ). X() (3.5.6) Because of the symmetry of the unperturbed orbits in the phase plane (3.5.6) can be rewritten as where ɛi(e β) = O( ɛ ) (3.5.7) I(E β) = 4 A ( X 9( X ) 3 ) d X (3.5.8) with A = X( T ). To have a periodic solution for (3.4.) we have to find an energy E such that I(E β) is equal to zero (see also [ ]). To find this constant energy E we rewrite I(E β) in where I(E β) = 4I (E β) ( 9 I (E β) I (E β) ) (3.5.9) I (E β) = A (E X β X ) 4 d X I (E β) = A (E X β X ) 3 4 d X. (3.5.)

48 3.5 Time-periodic solutions and a bifurcation analysis 39 X X Figure 3.: Phase portrait of the unperturbed equation (3.4.) with ɛ = µ = and λ >. It should be observed that the unperturbed equation (3.4.) with ɛ = has one equilibrium point ( ) which is center point. The phase portrait of the unperturbed equation (3.4.) is given in Figure 3.. It should be observed from (3.4.) that E. Putting Q(E β) = I (Eβ) (where I I (Eβ) and I are given by (3.5.)) it can be shown elementarily that lim E Q(E β) = and that Q ( β) = 3 >. There E is strong numerical evidence (see Figure 3.3 (g) and (h)) that Q is monotonically increasing. So we can conclude that there exists a unique nontrivial value for E such that I(E β) = or equivalently Q(E β) =. From these results it can be concluded 9 (see also for instance [[4] section 4. ]) that there exists a unique nontrivial stable time-periodic solution for (3.4.). The period T of this periodic solution is given by T = B A d X ( E X β X ) (3.5.) 4 where A = + β β + 4βE and B = the energy for which the periodic solution occurs. The case µ = and λ < β + β + 4βE in which E is Let T < be the period of a periodic solution and let c 3 be a constant in the first integral F (E ψ t ; ɛ) = constant for which a periodic solution exists. Consider F = c 3 for τ = and τ = T. Approximating F by F 5 (as given by (3.4.)) eliminating c 3 by subtraction we then obtain (using the fact that E() = E(T ) for a periodic solution) ( T ( ɛ ( X ) 9( X ) 4) ) d t = O( ɛ ) ɛ ( X(T ) X() ) ( X 9( X ) ) 3 d X = O( ɛ ). (3.5.)

49 4 Strongly nonlinear oscillators Because of the symmetry of the unperturbed orbits in phase plane (3.5.) can be rewritten as where ɛi(e γ) = O( ɛ ) (3.5.3) I(E γ) = 4 A ( X 9( X ) 3 ) d X (3.5.4) with A = X( T ). To have a periodic solution for (3.4.8) we have to find an energy E such that I(E γ) is equal to zero (see also [ ]). To find this energy E we rewrite I(E γ) in ( I(E γ) = 4I (E) 9 I ) (E γ) (3.5.5) I (E γ) where I (E γ) = A I (E γ) = A (E X + γ X 4 ) d X (E X + γ X 4 ) 3 d X. (3.5.6) It should be observed that the unperturbed equation (3.4.8) with ɛ = has three equilibrium points: a center point in ( ) and two saddles in (± γ ). The phase portrait of the unperturbed equation (3.4.8) is given in Figure 3.. For the peri- X γ γ X Figure 3.: Phase portrait of the unperturbed equation (3.4.8) with ɛ = µ = and λ <. odic (that is closed) orbits of the unperturbed equation (3.4.8) it can be deduced from (3.4.9) that E E max =. Putting Q(E γ) = I (Eγ) 4γ I (where I (Eγ) and I are given by (3.5.6)) it can be shown analytically that lim E Q(E γ) = lim E Q(E γ) = Q ( γ) = 3 Q > and ( γ) =. By using an 4γ 35γ E E 4γ adaptive Clenshaw-Curtis quadrature scheme Q(E γ) has been calculated numerically for different values of the parameter γ (or λ). These numerical results can

50 3.5 Time-periodic solutions and a bifurcation analysis 4 be found in Figure 3.3 (a)-(e). Using these numerical results we can try to find nontrivial values of E such that I(E γ) = or equivalently Q(E γ) = for a 9 given value of γ = λ >. The numerical results can be summarized as follows: 9 for λ < there are no nontrivial values of E such that Q = and 9 so there are no limit cycles; for λ = there are two coinciding nontrivial values of E such that Q = and so there is a semi-stable limit cycle; for < λ < there are two different nontrivial values of E such that Q = and so there are two limit cycles (one stable and one unstable); for < λ < there is exactly one nontrivial value of E such that Q = and 9 so there is exactly one (stable) limit cycle. It should be remarked that two limit cycles will occur in an extremely small interval for λ. For the asymptotics to be valid it should be noted that of course ɛ should tend to zero but also that ɛ should be much smaller than the length of this small interval. In the next sections small intervals will again occur and the aforementioned remark should be kept in mind. In Figure 3.4 a sketch of the appearance and disappearance of limit cycles is given for decreasing values of the parameter λ (and µ = ) The case µ > and λ arbitrary The case µ < and λ arbitrary can be treated similarly by simply replacing X by X. Let T < be the period of a periodic solution and let c 4 be a constant in a first integral F = constant for which a periodic solution exists. Consider F = c 4 for τ = and τ = T. Approximating F by F 7 (as given by (3.4.3)) and eliminating c 4 by subtraction we then obtain ( T ) ( ) ɛ G [] ηg [] d t = O( ɛ ) (3.5.7) or equivalently ( ) Z(T ) ( ) ɛ G [] ηg [] dz = O( ɛ ) ɛi(e ξ η) = O( ɛ ) (3.5.8) where Z() I(E ξ η) = B A G [] ( ηg [] ) dz (3.5.9) with A = Z() and B = Z( T ). To have a periodic solution for equation (3.4.7) we have to find an energy E such that I(E ξ η) is equal to zero (see also [ ]). To find this energy E we rewrite I(E ξ η) in I(E ξ η) = I (E ξ) ( η I (E ξ) I (E ξ) ) (3.5.) where I (E ξ) = B A I (E ξ) = B A ( E Z 3 Z3 ξz4) dz ( E Z 3 Z3 ξz4) 3 dz. (3.5.)

51 4 Strongly nonlinear oscillators Q Q E E (a) Zoom in for λ = (b) Zoom in for λ = Q Q E E (c) Zoom in for λ = (d) λ = 3 Q 38 Q E E (e) λ = (f) λ = 4 Q 4 Q 3 3 E E (g) λ = (h) λ = Figure 3.3: Plot of Q as function of E for µ = and for different values of λ.

52 3.5 Time-periodic solutions and a bifurcation analysis 43 λ > λ = λ < λ < λ < λ < : Stable : Unstable : Center Point : Saddle Point : Semistable Figure 3.4: Sketch of the appearance and disappearance of limit cycles for µ = and for decreasing values of the parameter λ. Throughout this section Q(E ξ) is equal to I (Eξ) I. The equilibrium points in the (Eξ) (Z Z ) phase plane of the unperturbed equation (3.4.7) with ɛ = are listed in Table 3. and the corresponding phase portraits of the unperturbed equation are given in Figure 3.5. To determine whether or not I(E ξ η) can be (nontrivially) equal to zero (or equivalently Q(E ξ) = ) we have to distinguish five cases: ξ < η ξ = < ξ < ξ = and ξ >. These five cases will be studied in the following five subsections. The case ξ < The unperturbed equation (3.4.7) with ɛ = and ξ < has only periodic orbits surrounding the center point ( ). For these periodic orbits the energy E satisfies: = E min E E max = z + 3 z3 + 4 ξz4 where z is ξ + ξ 4ξ. Putting Q(E ξ) = I (Eξ) (where I I (Eξ) and I are given by (3.5.)) it can be shown elementarily that lim E Emin Q = and that lim E Emax Q = P (ξ). P as function of ξ can be calculated numerically and is given in Figure 3.6. It can also be shown analytically that Q (E E min ξ) = 3 Q > and that (E E max ξ) = (see also Figure 3.7). For several values of ξ we have calculated Q(E ξ) numerically by using an adaptive recursive Simpson rule. The numerical results are presented in Figure 3.7. From Figure 3.7 it is clear that there are cases for which we can find two nontrivial values of E such that I(E ξ η) = or equivalently Q(E ξ) =. So we can conclude that η for (3.4.7) with ξ < at least two limit cycles can occur. It should be remarked that when two limit cycles are bifurcated out of the periodic orbits around the

53 44 Strongly nonlinear oscillators Z Z Z Z (a) ξ < (b) ξ = Z Z Z Z (c) < ξ < 9 (d) ξ = 9 Z Z Z Z (e) 9 < ξ < 4 (f) ξ = 4 Z Z (g) ξ > 4 Figure 3.5: Phase portrait of the unperturbed equation (3.4.7) with ɛ = for several values of ξ.

54 3.5 Time-periodic solutions and a bifurcation analysis 45 ξ Type and position of Equilibrium point(s) ( ξ < a center in () a saddle in ) ξ ξ 4ξ ( and a saddle in + ) ξ ξ 4ξ ξ = a center in () and a saddle ( in (-) < ξ < a center in () a center in ) 4 ξ ξ 4ξ ( and a saddle in + ) ξ ξ 4ξ ξ = a center in ( ) and a higher order singularity in ( ) 4 ξ > a center in ( ) 4 Table 3.: Type of equilibrium points of the unperturbed equation (3.4.7) with ɛ = in the (Z Z ) phase plane... P ξ Figure 3.6: Plot of P as function of ξ for ξ <. center point then these two limit cycles are very close each other. In fact the energy level of the stable periodic solution is only a little bit less then the energy level of the unstable periodic solution. The period T of the periodic solution(s) can be determined from dz dτ = Z = or equivalently from ( E Z 3 Z3 ) ξz4 (3.5.) T = B A dz (E where (3.5.3) Z 3 Z3 ξz4)

55 46 Strongly nonlinear oscillators Q Q E (a) ξ = E (b) Zoom in for ξ =. Q.4946 Q E E (c) ξ = (d) Zoom in for ξ =. Q.347 Q E E (e) ξ = (f) Zoom in for ξ = Figure 3.7: Plot of Q(E; ξ) as function of E for several values of ξ <.

56 3.5 Time-periodic solutions and a bifurcation analysis 47 ( + ) ξ ξ 4ξ A < B ( ) 6ξ ξ 4ξ + 3ξ + 3 4ξ and where A and B satisfy Z + 3 Z3 + 4 ξz4 = E in which E is the energy for which a periodic solution occurs. The case ξ = This case has already been studied in [4] and we refer the reader to this paper for detailed calculations. It has been shown in [4] that at most two limit cycles can bifurcated out of the periodic orbits surrounding the center point. A sketch of the appearance and disappearance of limit cycles for ξ = and decreasing values of η is given in Figure 3.8. : Stable : Unstable : Center Point : Saddle Point : Semistable Figure 3.8: Sketch of the appearance and disappearance of limit cycles for ξ = and for decreasing value of η. The case < ξ < 4 The unperturbed equation (3.4.7) with ɛ = has in this case three equilibrium ( points in the phase plane: a center point in σ = ( ) a center point in σ = ) ( ξ ξ 4ξ and a saddle point in σ 3 = + ) ξ ξ 4ξ (see also Table 3. and Figure 3.5). The maximum energy level for the periodic orbit inside the saddle loop connection (around the center point in σ and around the center point in σ ) and the minimum energy level for the periodic orbits outside the saddle loop connection are equal to E loop = z + 3 z3 + 4 ξz4 where z is + ξ ξ 4ξ. The minimum energy level E min[σ ] for periodic orbits surrounding the center point in σ is E min[σ ] = z + 3 z3 + 4 ξz4 where z is ξ ξ 4ξ. Putting Q(E ξ) = I (Eξ) I (Eξ) (where I and I are given by (3.5.)) it can be shown analytically that:

57 48 Strongly nonlinear oscillators (i) for the periodic orbits surrounding the center point in σ : lim Q(E ξ) = lim E Emin[σ ] E E loop Q(E ξ) = R σ (ξ) Q (E E min[σ ] ξ) = 3 Q and (E E loop ξ) = (ii) for the periodic orbits surrounding the center point in σ : lim Q(E ξ) = lim E Emin[σ ] E E loop Q(E ξ) = R σ (ξ) Q (E E min[σ ] ξ) = 3 Q and (E E loop ξ) = and (iii) for the periodic orbits outside the saddle loop connection: lim E Eloop Q(E ξ) = R σ3 (ξ) lim E Q(E ξ) = Q E (E loop ξ) = where R σi (ξ) for i = and 3 can be determined numerically and are given in Figure 3.9. Using an adaptive recursive Simpson rule Q(E ξ) has been calculated R 8 R R σ R σ3 4 R σ3 R σ 4 R σ R σ ξ ξ (a) R σi (ξ) for ξ 4. (b) Zoom in near ξ = 9. Figure 3.9: Plot of R as function of ξ. numerically for several values of ξ. Plots of Q are given in Figure 3. for ξ = in 9 Figure 3. for ξ = 7 and in Figure 3. for ξ =. From these figures it is obvious 9 7 that there are always non-trivial E-values such that I(E ξ η) = or equivalently Q(E ξ) =. Each non-trivial E-value corresponds to a limit cycle in the phase η plane. It also follows from these numerical calculations that at most two limit cycles can be bifurcated out of the periodic orbits surrounding σ. The same result also holds for the periodic orbits surrounding σ and for the periodic orbits outside the saddle loop. However out of all the periodic orbits at most five limit cycles can be bifurcated simultaneously. Numerical calculations give the following results: (i) for < ξ < at most three limit cycles can be bifurcated out of the periodic orbits (ii) for < ξ < at most five limit cycles can be bifurcated out of the periodic orbits and (iii) for < ξ < 4 of the periodic orbits. at most three limit cycles can be bifurcated out

58 3.5 Time-periodic solutions and a bifurcation analysis 49 Q Q E E (a) Surrounding σ (b) Zoom in surrounding σ Q Q E E.... (c) Surrounding σ (d) Zoom in surrounding σ Q 35 Q E (e) Outside saddle loop 3.5 E (f) Zoom in outside saddle loop Q Q σ Q σ 3 Q σ (g) Sketch for all E Figure 3.: Plot of Q(E; ξ) as function of E for ξ = 9.

59 5 Strongly nonlinear oscillators Q Q E E (a) Surrounding σ (b) Zoom in surrounding σ Q Q E E (c) Surrounding σ (d) Zoom in surrounding σ Q Q E E (e) Outside saddle loop (f) Zoom in outside saddle loop Q Q E E (g) For all (h) Zoom in for all Figure 3.: Plot of Q(E; ξ) as function of E for ξ = 9.

60 3.5 Time-periodic solutions and a bifurcation analysis Q Q E E (a) Surrounding σ (b) Zoom in surrounding σ Q.444 Q E E (c) Surrounding σ (d) Zoom in surrounding σ 5 Q Q E E (e) Outside saddle loop (f) Zoom in outside saddle loop Q Q σ Q σ 3 Q σ (g) Sketch for all E Figure 3.: Plot of Q(E; ξ) as function of E for ξ = 7 7.

61 5 Strongly nonlinear oscillators For ξ = and decreasing values of η and for ξ = and decreasing values of η 9 9 sketches of the appearance and disappearance of limit cycles are presented in Figure 3.3 and in Figure 3.4 respectively. The period T of a periodic solution can again : Stable : Unstable : Center Point : Saddle Point : Semistable Figure 3.3: Sketch of the appearance and disappearance of limit cycles for ξ = 9 and for decreasing values of η. be determined as is indicated in section 5... The case ξ = 4 The unperturbed equation (3.4.7) with ɛ = has in this case two equilibrium points in the phase plane: a center point in σ = ( ) and a higher order singularity in σ = ( )(see also Table 3. and Figure 3.5). The maximum energy level for the periodic orbits inside the loop connection (that is the orbit which starts in ( ) and ends in ( )) and the minimum energy level for the periodic orbits outside the loop connection are equal to E loop =. The minimum energy level for the periodic 3 orbits surrounding the center point in σ = ( ) is. Putting Q(E ξ) = I (Eξ) I (Eξ) (where I and I are given by (3.5.)) it can be shown analytically that: (i) for the periodic orbits inside the loop connection: lim E Q(E ) = 4 lim E Eloop Q(E ) = 4 Q ( ) = 3 Q and (E 4 9 E 4 E loop ) = and 4 (ii) for the periodic orbits outside the loop connection: lim E Eloop Q(E ) = lim E Q(E Q ) = (E 4 E loop ) =. 4 Again using an adaptive recursive Simpson rule Q(E ) has been calculated numerically. Plots of Q(E ) are given in Figure 3.5. From this figure it is clear 4 4

62 3.5 Time-periodic solutions and a bifurcation analysis 53 : Stable : Unstable : Semistable : Center Point : Saddle Point Figure 3.4: Sketch of the appearance and disappearance of limit cycles for ξ = 9 and for decreasing values of η. that there are always non-trivial E-values such that I(E ξ η) = or equivalently Q(E ξ) =. Each non-trivial E-value corresponds to a limit cycle in the phase η plane. It also follows from these numerical calculations that for ξ = at most three 4 limit cycles can be bifurcated out of the periodic orbits. More explicitly in this case: for < < three limit cycles will occur and for < or η η for > exactly one limit cycle will occurs. A sketch of the appearance η and disappearance of limit cycles for ξ = and for decreasing values of η is given in 4 Figure 3.6. The case ξ > 4 The unperturbed equation (3.4.7) with ɛ = has in this case only one equilibrium point in the phase plane: a center in () (see also Table 3. and Figure 3.5). The minimum and the maximum energy level for the periodic orbits are in this case and respectively. Putting Q(E ξ) = I (Eξ) (where I I (Eξ) and I are given by (3.5.)) it can be shown analytically that lim E Emin Q(E ξ) = lim E Q(E ξ) = and Q ( ξ) = 3. Again Q(E ξ) has been calculated numerically and plots of Q(E ξ) E are given in Figure 3.7 for different values of ξ. From this figure it is clear that there are always non-trivial E-values such that I(E η ξ) = or equivalently Q(E ξ) =. η It also follows from these numerical calculations that for < ξ < at 4 4 most three limit cycles and for ξ > at most one limit cycle can be 4 bifurcated out of the periodic orbits. A sketch of the appearance and disappearance of limit cycles for < ξ < and for decreasing values of η is given in 4 4

63 54 Strongly nonlinear oscillators.44 Q.4455 Q E E (a) Inside loop (b) Zoom in inside loop Q Q (c) Outside loop E E (d) Zoom in outside loop Q Q E (e) For all combined E (f) Zoom in Figure 3.5: Plot of Q(E; ξ) as function of E for ξ = 4.

64 3.6 Conclusions and remarks 55 : Stable : Unstable : Semistable : Center Point : Higher order singularity if ξ = 4 Figure 3.6: Sketch of the appearance and disappearance of limit cycles for 4 ξ < and for decreasing values of η. Figure Conclusions and remarks In this chapter it has been shown that the perturbation method based on integrating factors can be used efficiently to approximate first integrals for strongly nonlinear oscillator equations. In section 3. (and 3.3) of this chapter an asymptotic justification of the presented perturbation method has been given. It has been shown how the existence and stability of time-periodic solutions can be deduced from the approximations of the first integrals. In section 3.4 it has been shown explicitly how approximations of first integrals can be constructed for the generalized Rayleigh oscillator equation Ẍ + 9X + µx + λx 3 = ɛ(ẋ Ẋ3 ) (3.6.4) where µ and λ are parameters and where ɛ is a small parameter satisfying < ɛ. In section 3.5 it has been shown how the existence of time-periodic solutions of (3.6.4) can be determined from the approximations of first integrals. The following results for the generalized Rayleigh oscillator equation (3.6.4) have been obtained in this chapter: (partially based on strong numerical evidence) (i) for µ = and λ and for µ and 9λ µ > : exactly one limit cycle is bifurcated out of the periodic orbits (of the unperturbed equation (3.6.4) with ɛ = ).

65 56 Strongly nonlinear oscillators.8 Q.46 Q E.4 E (a) ξ =.5 (b) Zoom in for ξ =.5.8 Q.47 Q E.4 E (c) ξ =.54 (d) Zoom in for ξ =.54.8 Q.478 Q E E (e) ξ =.56 (f) Zoom in for ξ =.56 Q Q E E (g) ξ =.3 (h) ξ = Figure 3.7: Plot of Q(E; ξ) as function of E for ξ > 4.

66 3.6 Conclusions and remarks 57 (ii) for λ < and for µ and λ = : at most two limit cycles can be bifurcated out of the periodic orbits. (iii) for µ and < 9λ < µ and for µ and < 9λ : at most three limit cycles can be bifurcated out of the µ 4 periodic orbits. (iv) for µ and < 9λ µ < : at most five limit cycles can be bifurcated out of the periodic orbits.

67 58 Strongly nonlinear oscillators

68 Chapter 4 On Approximations of First Integrals for a System of Weakly Nonlinear Coupled Harmonic Oscillators Abstract. In this chapter a system of weakly nonlinear coupled harmonic oscillators will be studied. It will be shown that the recently developed perturbation method based on integrating vectors can be used to approximate first integrals and periodic solutions. To show how this perturbation method works the method is applied to a system of weakly nonlinear coupled harmonic oscillators with :3 and 3: internal resonances. Not only approximations of first integrals will be given but it will also be shown how in a rather efficient way the existence and stability of time-periodic solutions can be obtained from these approximations. In addition some bifurcation diagrams for a set of values of the parameters will be presented. 4. Introduction In [ ] a perturbation method based on integrating factors and vectors has been presented for regularly perturbed systems of ordinary differential equations (ODEs). When approximations of integrating vectors have been obtained an approximation of a first integral can be given. Also an error-estimate for this approximation of a first integral can be given on a time-scale. It has also been shown in [ ] how in a rather efficient way the existence and stability of timeperiodic solutions can be obtained from these approximations for the first integrals. In this chapter it will be shown how the perturbation method can be applied to systems of weakly nonlinear coupled harmonic oscillators. In the literature many This chapter is a revised version of [43] On Approximations of First Integrals for a System of Weakly Nonlinear Coupled Harmonic Oscillators Nonlinear Dynamics 3:

69 6 A Weakly Nonlinear Coupled Harmonic Oscillators mathematical models have been considered describing the dynamics of systems with two degrees of freedom. Verros and Natsiavas [39] considered the dynamics of symmetric self-excited oscillators with an one-to-two internal resonance. Natsiavas [7] studied also the free vibrations of a weakly nonlinear oscillator using a multiple time scales perturbation method. Haaker and van der Burgh [6] used an averaging method to study nonlinear rotational galloping for two mechanically coupled seesaw oscillators in steady cross-flow. Mitsi Natsiavas and Tsiafis [6] considered a class of weakly nonlinear oscillators with symmetric restoring forces. The weakly nonlinear resonant response of systems with multiple degrees of freedom to simple harmonic excitations has been extensively studied by Nayfeh and Mook [8]. Bajaj Chang and Johnson [] and others have studied forced weakly nonlinear oscillations with two degrees of freedom as model for autoparametric vibration absorbers with resonant excitations. In this chapter the recently developed perturbation method based on integrating factors and vectors will be used to approximate first integrals and periodic solutions for the following weakly nonlinear system Ẍ + ( ω + ɛδ ) ] X = ɛ [ a Ẋ + a Ẏ + a Ẋ a ẊẎ + a Ẏ a 3 Ẏ 3 Ÿ + ( ω + ɛδ ) ] Y = ɛ [ b Ẋ + b Ẏ + b Ẋ b ẊẎ b Ẏ b 3 Ẏ 3 (4..) where X = X(t) Y = Y (t) and where ɛ is a small parameter satisfying < ɛ. The constants a ij b ij have the following properties: a a a b b b 3 are positive and a a a 3 b b b have the same sign. As follows from measurements of aerodynamic coefficients in a wind-tunnel these signs are relevant for the description of a galloping phenomenon. The dot represents differentiation with respect to t. The constants δ δ are detuning parameters. We consider in this chapter the : 3 and 3 : internal resonances that is ω : ω = : 3 or 3 :. The frequencies ω and ω are assumed to be constants. In this chapter not only asymptotic approximations of first integrals are constructed but also asymptotic approximations of periodic solutions. The presented results include existence uniqueness and stability properties of the periodic solutions. In [3] van der Beek uses (4..) without detuning parameters δ δ as mathematical model to describe flow-induced vibrations of two weakly non-linear coupled harmonic oscillators in a uniform windfield. The model problem originates from the phenomenon of galloping of overhead transmission lines on which ice has accreted. These conductors can become aerodynamically unstable resulting in large amplitude oscillations with low frequencies. The oscillator consists of a rigid cylinder with small ridge and a number of springs mounted in a frame. The oscillator is constructed in such a way that the cylinder-spring system has two degrees of freedom i.e. oscillation in the direction and oscillations perpendicular to the windfield. A more detailed description is given in [3] whereas a short summary is included in Appendix A. The internal resonances that will be studied in this chapter have not been studied in [3]. This chapter is organized as follows. In section 4. of this chapter the perturbation method based on integrating vectors and an asymptotic theory will be given briefly. It will be shown in section 4.3 of this chapter how approximations of first integrals can be constructed for systems of weakly nonlinear coupled harmonic oscillators. In section 4.4 it will be shown how the existence and stability of time-periodic solutions

70 4. Integrating vectors and an asymptotic theory 6 can be obtained. We will also present some bifurcation diagrams for a set of values of the parameters. Finally in section 4.5 of this chapter some conclusions will be drawn and some remarks will be made. 4. Integrating vectors and an asymptotic theory In this section we briefly outline the perturbation method based on integrating vectors as given in [ ]. We consider the following system of n first order ODEs dy dt = f(y t; ɛ) (4..) where ɛ is a small parameter and where the function f has the form f(y t; ɛ) = f (y t) + ɛf (y t). An integrating vector µ = µ(y t; ɛ) for system (4..) has to satisfy µ i y j = µ j y i i < j n µ t = (µ f). (4..) Assume that µ can be expanded in a power series in ɛ that is µ(y t; ɛ) = µ (y t) + ɛµ (y t) ɛ m µ m (y t) We determine an integrating vector up to O(ɛ m ). An approximation F app of F in the first integral F = constant can be obtained from: F app = µ + ɛµ ɛ m µ m ) ] (4..3) F app = [(µ t + ɛµ ɛ m µ m f where the * indicates that terms of order ɛ m+ and higher have been neglected. Then we obtain F app (y t; ɛ) = F (y t) + ɛf (y t) ɛ m F m (y t). It should be observed that an approximation up to O(ɛ m ) of an integrating vector µ has been used to obtain an exact ODE up to O(ɛ m+ ) that is df app dt = ) ] [(µ + ɛ µ ɛ m µ m f = ɛ m+ R m+ (y t µ... µ m ; ɛ) (4..4) where the ** indicates that only terms of order ɛ m+ and higher are included. How well F app approximates F (y t; ɛ) = constant can be determined from (4..4) that is error estimates can be given on time-scales depending on the boundedness properties of R m Approximations of First Integrals In this section we will show how the perturbation method based on integrating vectors can be applied to approximate first integrals for a system of weakly nonlinear

71 6 A Weakly Nonlinear Coupled Harmonic Oscillators coupled harmonic oscillators. In the first part of this section we will consider system (4..) with a :3 internal resonance and the second part with a 3: internal resonance. First of all we remark that initial value problems for system (4..) (or for the equivalent systems (4.3.) or (4.3.3)) are well-posed on timescales of order (for proof we refer the reader to for instance Verhulst [38]). We consider an n- ɛ dimensional system with a small parameter ɛ. When ɛ = the system is integrable. As is well-known a near-integrable system may be integrable or not but we assume integrability for ɛ >. When existence of solutions for a system of n first order ODEs has been established it is well-known that this systems of n first order ODEs has n and can not have more than n functionally independent first integrals. A complete proof for this statement can be found in Forsyth [[3] p ]. Similar remarks on the existence of n functionally independent first integrals can also be found in for instance [ 4 8 9]). It should be observed that these n functionally independent first integrals exist on a time-scale of order. Since the right-hand sides of (4..) (and of (4.3.) and (4.3.3)) are analytic in ɛ it can also be deduced from the proof in [3] that there are n functionally independent first integrals which are analytic in ɛ (implying that also the corresponding integrating factors/vectors exist and are analytic in ɛ). In this section we will show how approximations of first integrals can be obtained by expanding the integrating factors/vectors in power series in ɛ The :3 internal resonance case Consider the mathematical model which describes the flow-induced vibrations of an oscillator with two degrees of freedom in a uniform windfield with a :3 internal resonance ] Ẍ + ( + ɛδ ) X = ɛ [ a Ẋ + a Ẏ + a Ẋ a ẊẎ + a Ẏ a 3 Ẏ 3 ] Ÿ + (9 + ɛδ ) Y = ɛ [ b Ẋ + b Ẏ + b Ẋ b ẊẎ b Ẏ b 3 Ẏ 3. (4.3.) To analyze system (4.3.) the equations are first written as a system of first order ODEs. Let X = X Ẋ = X Y = X 3 Ẏ = X 4 from (4.3.) we then obtain X = X X = X + ɛ [ δ X a X + a X 4 X 3 = X 4 +a X a X X 4 + a X 4 a 3X 3 4 ] (4.3.) X 4 = 9X 3 + ɛ [ δ X 3 b X + b X 4 +b X b X X 4 b X 4 b 3 X 3 4 ].

72 4.3 Approximations of First Integrals 63 By using the transformation X = r cos(θ ) X = r sin(θ ) X 3 = r cos(3θ ) and X 4 = 3r sin(3θ ) system (4.3.) then becomes dr dt = ɛg (r r θ θ ) = f (r r θ θ ) dθ dt = + ɛg (r r θ θ ) = f (r r θ θ ) dr dt = ɛg 3 (r r θ θ ) = f 3 (r r θ θ ) dθ dt = + ɛg 4 (r r θ θ ) = f 4 (r r θ θ ) (4.3.3) where g g g 3 and g 4 are given in Appendix B formula (4.7.3). Multiplying the first the second the third and the fourth equation in (4.3.3) by µ µ µ 3 and µ 4 respectively it follows from (4..) that the integrating factors µ µ µ 3 and µ 4 have to satisfy (µ i = µ i (r r θ θ t) for i = 3 and 4) µ θ = µ r µ r = µ 3 r µ r = µ 3 θ µ θ = µ 4 r µ θ = µ 4 θ µ 3 θ = µ 4 r µ t µ t = r (µ f + µ f + µ 3 f 3 + µ 4 f 4 ) = θ (µ f + µ f + µ 3 f 3 + µ 4 f 4 ) (4.3.4) µ 3 t = r (µ f + µ f + µ 3 f 3 + µ 4 f 4 ) µ 4 t = θ (µ f + µ f + µ 3 f 3 + µ 4 f 4 ). Expanding µ µ µ 3 and µ 4 in powers series in ɛ that is µ i (r r θ θ t)= µ i (r r θ θ t)+ɛµ i (r r θ θ t)+... (for i= 3 and 4) substituting f f f 3 f 4 and the expansions for µ µ µ 3 and µ 4 into (4.3.4) and by taking together terms of equal powers in ɛ we finally obtain the O(ɛ )-problem µ θ = µ r µ r = µ 3 r µ r = µ 3 θ µ θ = µ 4 r µ t µ 3 t µ θ = µ 4 θ = r ( µ µ 4 ) µ t = r ( µ µ 4 ) µ 4 t µ 3 θ = µ 4 r = θ ( µ µ 4 ) = θ ( µ µ 4 ) (4.3.5)

73 64 A Weakly Nonlinear Coupled Harmonic Oscillators and for n the O(ɛ n )-problems µ n θ = µ n r µ n r = µ 3n r µ n r = µ 3n θ µ n θ = µ 4n r µ n θ = µ 4n θ µ 3n θ = µ 4n r µ n t µ n t = r (µ n g + µ n g µ n + µ 3n g 3 + µ 4n g 4 µ 4n ) = θ (µ n g + µ n g µ n + µ 3n g 3 + µ 4n g 4 µ 4n ) (4.3.6) µ 3n t = r (µ n g + µ n g µ n + µ 3n g 3 + µ 4n g 4 µ 4n ) µ 4n t = θ (µ n g + µ n g µ n + µ 3n g 3 + µ 4n g 4 µ 4n ). The O(ɛ )-problem (4.3.5) can easily be solved yielding µ = h (r r θ +t θ + t) µ = h (r r θ +t θ +t) µ 3 = h 3 (r r θ +t θ +t) µ 4 = h 4 (r r θ + t θ + t) with h h 3 θ = h 4 r θ = h r h r = h 3 r h θ = h 4 r h r = h 3 θ h θ = h 4 θ. The functions h h h 3 and h 4 are still arbitrary and will now be chosen as simple as possible. First we choose h = and h = h 3 = h 4 = or equivalently (µ µ µ 3 µ 4 ) = ( ). Then the O(ɛ )-problem (4.3.6) can also readily be solved yielding µ = a t + 4 δ cos(θ ) + 4 a sin(θ ) 3 a r cos(θ ) + 6 a r cos(3θ ) + a r cos(3θ ) 3 a r cos(3θ + θ ) 3 4 a r cos(3θ θ ) +h (r r θ + t θ + t) µ = δ r sin(θ ) + a cos(θ ) 3 4 a r cos( θ + 3θ ) 3 8 a r cos(θ + 3θ ) a r sin(θ ) 4 a r sin(3θ ) + 3 a r r sin(3θ + θ ) 3 a r r sin(3θ θ ) + 9 a r sin(θ ) 9 8 a r sin(θ + 6θ ) 9 a r sin( θ + 6θ ) a 3r 3 cos( θ + 3θ ) a 3r 3 cos(θ + 3θ ) 7 64 a 3r 3 cos(9θ θ ) 7 8 a 3r 3 cos(9θ + θ ) +h (r r θ + t θ + t)

74 4.3 Approximations of First Integrals 65 µ 3 = 3 4 a sin( θ + 3θ ) 3 8 a sin(θ + 3θ ) + a r cos(3θ ) 3 a r cos(3θ + θ ) 3 4 a r cos(3θ θ ) 9a r cos(θ ) a r cos(θ + 6θ ) 9 a r cos(θ 6θ ) 43 6 a 3r sin( θ + 3θ ) a 3r sin(θ + 3θ ) a 3r sin(9θ θ ) 8 8 a 3r sin(9θ + θ ) +h 3 (r r θ + t θ + t) µ 4 = 9 4 a r cos( θ + 3θ 3 ) 9 8 a r cos(θ + 3θ 3 ) 3 a r r sin(3θ ) + 9 a r r sin(3θ + θ ) a r r sin(3θ θ ) 7 4 a r sin(θ + 6θ ) + 7 a r sin( θ + 6θ ) 43 6 a 3r 3 cos( θ + 3θ ) a 3r 3 cos(θ + 3θ ) (4.3.7) a 3r 3 cos(9θ θ ) 43 8 a 3r 3 cos(9θ + θ ) +h 4 (r r θ + t θ + t) where h h h 3 and h 4 have to satisfy h h r = h 3 θ h θ = h 4 θ h 3 θ = h 4 r θ = h r h r = h 3 r h θ = h 4 r. The functions h h h 3 and h 4 are still arbitrary and will now be chosen as simple as possible: h = h = h 3 = h 4 =. The O(ɛ n )-problems with n can also be solved. By using (4..3) and the approximation (+ɛµ ɛµ ɛ µ 3 ɛ µ 4 ) for the integrating vector (µ µ µ 3 µ 4 ) we can construct an approximation F of a first integral F = constant yielding where F = r + ɛ F (4.3.8) F = a r t + 4 δ r cos( θ ) + 4 a r sin( θ ) 3 4 a r cos(θ ) + a r cos(3 θ ) + a r r cos(3 θ ) 3 a r r cos(3 θ + θ ) 3 4 a r r cos( 3 θ + θ ) 3 4 a r sin(θ 3 θ ) 3 8 a r sin(θ + 3 θ ) 9 a r cos(θ ) a r cos(θ + 6 θ ) 9 a r cos(θ 6 θ ) a 3r 3 sin(θ 3 θ )

75 66 A Weakly Nonlinear Coupled Harmonic Oscillators a 3r 3 sin(θ + 3 θ ) 7 64 a 3r 3 sin( 9 θ + θ ) 7 8 a 3r 3 sin(9 θ + θ ). (4.3.9) How well F approximates F in a first integral F = constant follows from (4..4). In this case we have df dt = [f + ɛµ f + ɛµ f + ɛµ 3 f 3 + ɛµ 4 f 4 ] = ɛ R (r θ r θ ) (4.3.) where f f f 3 f 4 are given by (4.3.3) and µ µ µ 3 µ 4 are given by (4.3.7) respectively. From the existence and uniqueness theorems for ODEs we know that an initial-value problem for (4.3.) is well-posed on a time-scale of order ɛ. This implies that also an initial-value problem for system (4.3.3) is well-posed on this time-scale. From (4.3.3) it then follows on this time-scale that if r () r () are bounded then r (t) r (t) are bounded and θ (t) θ (t) are bounded by constants plus t. Since R c + c t on a time scale of order ɛ where c c are constants it follows from (4.3.) that t F (r (t) θ (t) r (t) θ (t) t; ɛ) = constant + ɛ R (r (s) θ (s) r (s) θ (s))ds This implies that and F (r (t) θ (t) r (t) θ (t) t; ɛ) = constant + O(ɛ t) + O(ɛ t ). F (r (t) θ (t) r (t) θ (t) t; ɛ) = constant + O(ɛ ) for t T < F (r (t) θ (t) r (t) θ (t) t; ɛ) = constant + O(ɛ) for t L ɛ (4.3.) (4.3.) where T and L are ɛ-independent constants. By putting (µ µ µ 3 µ 4 ) = ( ) or ( ) or ( ) we can construct a second a third and a fourth (functionally independent) approximation F F 3 and F 4 of a first integral F = constant. After some elementary calculations we then obtain F = (θ + t) + ɛf F 3 = r + ɛf 3 (4.3.3) F 4 = (θ + t) + ɛf 4

76 4.3 Approximations of First Integrals 67 where F F 3 and F 4 are given in Appendix B formula (4.7.4). How well F F 3 and F 4 (as given by (4.3.3)) approximate F in a first integral F = constant can be determined similar to (4.3.)-(4.3.). It can be shown that and F (r (t) θ (t) r (t) θ (t) t; ɛ) = constant + O(ɛ ) for t T < F (r (t) θ (t) r (t) θ (t) t; ɛ) = constant + O(ɛ) for t L ɛ where T and L are ɛ-independent constants and F 3 (r (t) θ (t) r (t) θ (t) t; ɛ) = constant + O(ɛ ) for t T 3 < F 3 (r (t) θ (t) r (t) θ (t) t; ɛ) = constant + O(ɛ) for t L 3 ɛ where T 3 and L 3 are ɛ-independent constants and and F 4 (r (t) θ (t) r (t) θ (t) t; ɛ) = constant + O(ɛ ) for t T 4 < F 4 (r (t) θ (t) r (t) θ (t) t; ɛ) = constant + O(ɛ) for t L 4 ɛ (4.3.4) (4.3.5) where T 4 and L 4 are ɛ-independent constants The 3: internal resonance case We consider in this subsection ] Ẍ + (9 + ɛδ ) X = ɛ [ a Ẋ + a Ẏ + a Ẋ a ẊẎ + a Ẏ a 3 Ẏ 3 ] Ÿ + ( + ɛδ ) Y = ɛ [ b Ẋ + b Ẏ + b Ẋ b ẊẎ b Ẏ b 3 Ẏ 3. (4.3.6) (4.3.7) Putting X = X Ẋ = X Y = X 3 and Ẏ = X 4 it follows from (4.3.7) that X = X X = 9X + ɛ [ δ X a X + a X 4 X 3 = X 4 +a X a X X 4 + a X 4 a 3X 3 4 ] (4.3.8) X 4 = X 3 + ɛ [ δ X 3 b X + b X 4 +b X b X X 4 b X 4 b 3 X 3 4].

77 68 A Weakly Nonlinear Coupled Harmonic Oscillators By using the transformation X = r cos(3θ ) X = 3r sin(3θ ) X 3 = r cos(θ ) and X 4 = r sin(θ ) system (4.3.8) then becomes dr dt = ɛh (r r θ θ ) dθ dt = + ɛh (r r θ θ ) dr dt = ɛh 3 (r r θ θ ) dθ dt = + ɛh 4 (r r θ θ ) (4.3.9) where h h h 3 and h 4 are given in Appendix C formula (4.8.6). In a similar way as in subsection (3.) we can obtain after some elementary calculations approximations G G G 3 and G 4 of first integrals for system (4.3.9): G = r + ɛg G = (θ + t) + ɛg G 3 = r + ɛg 3 G 4 = (θ + t) + ɛg 4 (4.3.) where G G G 3 and G 4 are given explicitly in Appendix C formula (4.8.7). How well G i (i = 3 or 4) approximates G in a first integral G = constant follows from (4..4) (see also (4.3.)-(4.3.)). It can be shown that for i = 3 and 4 and G i (r (t) θ (t) r (t) θ (t) t; ɛ) = constant + O(ɛ ) for t T i < G i (r (t) θ (t) r (t) θ (t) t; ɛ) = constant + O(ɛ) for t L i ɛ where T i and L i for i = 3 and 4 are ɛ-independent constants. (4.3.) 4.4 Approximations for time-periodic solutions and analysis of bifurcations In section 4.3 we constructed asymptotic approximations of first integrals. In this section we will show how the existence the stability and the approximations of non-trivial time-periodic solutions can be determined from these asymptotic approximations of first integrals. We will also give some bifurcation diagrams for a set of values of the parameters.

78 4.4 Approximations for time-periodic solutions and analysis of bifurcations The :3 internal resonance case The asymptotic approximations (4.3.3) for the first integrals of the weakly nonlinear coupled harmonic oscillators with a :3 internal resonance can be used to determine the existence and stability of time-periodic solutions. Let T < be the period of a periodic solution and let c be a constant in the first integrals F (r θ r θ t ; ɛ) = constant for which a periodic solution exists. Consider F = c for t = nt and t = (n )T with n N + then F (r (nt ) θ (nt ) r (nt ) θ (nt ) nt ; ɛ) = c (4.4.) F (r ((n )T ) θ ((n )T ) r ((n )T ) θ ((n )T ) (n )T ; ɛ) = c. For the autonomous system (4.3.) we may assume that θ () = α and θ () = β where α and β are to be determined later. From (4.3.3) it follows that r (nt ) = r ((n )T ) + O(ɛ) θ (nt ) = θ ((n )T ) T + O(ɛ) r (nt ) = r ((n )T ) + O(ɛ) θ (nt ) = θ ((n )T ) T + O(ɛ). (4.4.) Approximating F by F (given by (4.3.8)) eliminating c from (4.4.) by subtraction and using (4.4.) we obtain ( r (nt ) = ) ɛt a r ((n )T ) + O(ɛ t) (4.4.3) on a time scale of order. Since a ɛ > we can see from (4.4.3) that r (nt ) decreases for increasing n. Hence the only possible candidate for a periodic solution is r. However for r = the approximation F in (4.3.3) is not valid. From (4.3.) it can readily be seen that the only resonance term in the right hand side of the equation for X is ɛa Ẋ and this is a damping term. So if X() is O(ɛ) then X(t) will be at most O(ɛ) for t >. So we only have to study ] Ÿ + (9 + ɛδ ) Y = ɛ [b Ẏ b Ẏ b 3 Ẏ 3 (4.4.4) when we are interested in periodic solutions. Approximations of first integrals can be obtained by taking r = in F 3 and F 4 (see (4.3.3)) yielding F 3 = r + ɛ [ 36 δ r cos(6 θ ) b r t b 3r 3 t b r sin(6 θ ) cos(3 θ )b r b r cos(9 θ ) b 3r 3 sin(6 θ ) 3 ] 3 b 3r 3 sin( θ ) (4.4.5)

79 7 A Weakly Nonlinear Coupled Harmonic Oscillators F 4 = θ + t + ɛ [ 8 δ t b r sin(3 θ ) + 36 b r sin(9 θ ) + 8 b 3r cos(6 θ ) 3 b 3r cos( θ ) 8 δ sin(6 θ ) b 36 cos(6 θ ) ]. (4.4.6) Let again T < be the period of a periodic solution and let c be a constant in a first integral F (r θ r θ t; ɛ) = contant for which a periodic solution exists. Consider F = c for t = nt and t = (n )T with n N + then F (r (nt ) θ (nt ) r (nt ) θ (nt ) nt ; ɛ) = c (4.4.7) F (r ((n )T ) θ ((n )T ) r ((n )T ) θ ((n )T ) (n )T ; ɛ) = c. Approximating F by F 3 (given by (4.4.5)) eliminating c from (4.4.7) by subtraction and using (4.4.) we obtain r (nt ) = r ((n )T ) + ɛt ( b r ((n )T ) 7 8 b 3r ((n )T ) 3) + O(ɛ t) (4.4.8) on a time scale of order. In fact (4.4.8) defines a map Q : r ɛ Q(r ) r n = Q(r n ) with r n = r (nt ). We define a new map P by neglecting the term of O(ɛ t) in (4.4.8). That is P : r P ( r ) r n = P ( r n ) with r n = r (nt ). It will be shown that for r > : ( (i) If r r = O(ɛ) for ɛ then r n r n = O(ɛ) for n = O ɛ ) that is for n ɛ and ɛ r n and r n remain ɛ-close. (ii) The map P has a unique hyperbolic fixed point r = b 3 3b 3 which is asymptotically stable. (iii) There exists an ɛ > such that for all < ɛ ɛ the map Q has a unique b hyperbolic fixed point r = 3 3b 3 + O(ɛ) with the same stability property as the fixed point r = b 3 3b 3 of the map P. P roof of (i): From r r = O(ɛ) for ɛ it follows that there exists a positive constant M such that r r = M ɛ. We have r n r n = P (r n ) P ( r n ) + O(ɛ n) P (r n ) P ( r n ) + M ɛ n L r n r n + M ɛ n (4.4.9) where M and L are positive constants with L = + ɛ M and M a positive constant. So we have r n r n ( + ɛm ) r n r n + M ɛ n... ɛ(m + ɛn M )e ɛnm (4.4.)

80 4.4 Approximations for time-periodic solutions and analysis of bifurcations 7 and so for n = O( ɛ ) we conclude that r n r n = O(ɛ). P roof of (ii): The fixed points of the map P follow from r n = P ( r n ) for n or equivalent from r = r + ɛt ( b r 7b 8 3 r ) 3 r (b 7b 4 3 r ) =. For r > we have a unique fixed point r = b 3 3b 3. The fixed point of the map P is hyperbolic if the linearized map around this fixed point has no eigenvalues of unit modulus. Let DP be this linearized map then DP = ɛt b. Since < ɛ and b > we have λ < and so the fixed point is hyperbolic and stable. P roof of (iii): For the proof of (iii) we refer to [33] for a similar proof. So far we can conclude that there exists an asymptotically stable nontrivial T - periodic solution for system (4.3.). We can conclude that the periodic solution for system (4.3.) is a combination of a trivial periodic solution in X-direction (that is r ) and a nontrivial periodic solution in Y -direction. It has been shown that the nontrivial time-periodic solution of the weakly nonlinear coupled harmonic oscillators with a :3 internal resonance can be determined from the first integrals (4.3.8) and (4.3.3) yielding X(t) and Y (t) = A cos(3θ (t)) where A = b 3 3b 3 and where θ (t) can be approximated from (4.3.3) or (4.3.6) by θ () ( + ɛ δ 8) t The 3: internal resonance case The four functionally independent asymptotic approximations (4.3.) for first integrals of system (4.3.7) can be used to determine the existence and stability of non-trivial time-periodic solutions for this system. Let T < be the period of a periodic solution and let c i (for i = 3 and 4) be constants in the first integrals G(r θ r θ t ; ɛ) = constant for which a periodic solution exists. Approximate G by G i (as given by (4.3.)) and consider G i + O(ɛ t) = c i i = 3 4 for t = nt and t = (n )T with n N + then for i = 3 and 4 we have G i (r (nt ) θ (nt ) r (nt ) θ (nt ) nt ; ɛ) + O(ɛ t) = c i G i (r ((n )T ) θ ((n )T ) r ((n )T ) θ ((n )T ) (n )T ; ɛ) +O(ɛ t) = c i (4.4.) where G i for i = 3 and 4 are given explicitly in Appendix C by formula (4.8.7). By eliminating the constants c i for i = 3 and 4 from (4.4.) by

81 7 A Weakly Nonlinear Coupled Harmonic Oscillators simple subtractions we obtain r (nt ) = r ((n )T ) + ɛt ( a r ((n )T ) θ (nt ) = + 4 a 3r ((n )T ) 3 cos(3θ ((n )T ) 3θ ((n )T )) ) + O(ɛ t) θ ((n )T ) T + ɛ ( 8 δ T ) 7 a 3T r ((n )T ) 3 r ((n )T ) sin(3θ ((n )T ) 3θ ((n )T )) (4.4.) + O(ɛ t) r (nt ) = r ((n )T ) + ɛt ( b r ((n )T ) 3 8 b 3r ((n )T ) 3) + O(ɛ t) θ (nt ) = θ ((n )T ) ɛ δ T + O(ɛ t). By letting ψ = θ θ we then obtain r (nt ) = r ((n )T ) + ɛt ( a r ((n )T ) + 4 a 3r ((n )T ) 3 cos(3ψ ((n )T )) ) + O(ɛ t) r (nt ) = r ((n )T ) + ɛt ( b r ((n )T ) 3 8 b 3r ((n )T ) 3) + O(ɛ t) ψ(nt ) = ψ ((n )T ) + ɛt (( δ δ ) 8 7 a 3 r ((n )T ) 3 r ((n )T ) sin(3ψ ((n )T )) ) + O(ɛ t). (4.4.3) In fact (4.4.3) defines a map which we will use to determine the nontrivial periodic solution(s) of system (4.3.7). First it should be remarked that the trivial periodic solution of system (4.3.7) (that is X(t) and Y (t) ) is unstable. This can readily be deduced from (4.3.7) or (4.3.8) by linearizing the system around the trivial solution. Since the only resonant term in the equation for X is ɛa 3 Ẏ 3 (see (4.3.7)) it is obvious that there can not be a nontrivial periodic solution for which X(t) unless a 3 =. As in section 4.4. (see also (4.4.8)) we can now study the map as defined by (4.4.3). A completely similar analysis as given in section 4.4. then yields that the map as defined by (4.4.3) has a unique stable nontrivial hyperbolic fixed point (r r ψ) which up to O(ɛ) is equal to (A A θ ) where 3b3 b A = b a 3 7b 3 cos(3θ ) = a A a 3 A 3 36δ +a A = 3 3b b 3 δ = δ 8 δ and where θ is given by and sin(3θ ) = 7δA a 3. From this it follows that the system of A 3 weakly nonlinear coupled harmonic oscillators with a 3: internal resonance has a nontrivial periodic solution X(t) = A cos(3θ (t)) and Y (t) = A cos(θ (t)) where θ (t) and θ (t) can be approximated from (4.3.9) or (4.3.) by θ () ( + ɛ(δ + 8 δ ))t and θ () ( + ɛ( δ ))t respectively with θ () θ () = θ.

82 4.4 Approximations for time-periodic solutions and analysis of bifurcations Analysis of Bifurcations In the weakly nonlinear system (4..) the coefficients a ij and b ij depend on the quasi-static drag and lift forces acting on a conductor in uniform windfield. These quasi-static forces C D (α) and C L (α) where α is the angle between the virtual windvelocity and the axis of symmetry of the conductor can be measured in a windtunnel. According to the Den Hartog criterion (a linear instability criterion for the equilibrium position) galloping may set in if C D (α) + C α L(α) < for α = α s where α s is the static angle of attack that is α s is the angle between the direction of the uniform windflow (in this case the X-direction) and the axis of symmetry of the conductor. Typical results from windtunnel measurements for a certain range of values α are:(see also [6 3 3]) C D (α) = C D C L (α) = C L (α α ) + C L3 (α α ) 3 where C D C L and C L3 are constants and where α is usually the angle of attack for which galloping sets in and for which C D (α)+ C α L(α) is as negative as possible. In this chapter we will use the parameter values C D = C L = 3 and C L3 = 6 (see also [3]). Furthermore we define ᾱ s = α s α. How the coefficients a ij and b ij depend on ᾱ s is given in [3] and in appendix A. Generally it is assumed that for ᾱ s = the largest oscillation amplitudes due to galloping occur. In this section we will show how this assumption can be verified. It should be remarked that in order to have galloping b = (C D + C L + 3 C L3 ᾱs) has to be positive implying that ᾱ s should satisfy in this case : ᾱ s < 6 5. The :3 internal resonance case Since a is independent of ᾱ s it follows from (4.4.3) that X(t) tends to zero for increasing time. And from (4.4.8) it follows that r = is unstable for ᾱ s < 6 5. So the trivial periodic solution is unstable for ᾱ s < 6 5. For the nontrivial periodic solution X(t) = and Y (t) = A cos(3θ (t)) it follows easily from proof (ii) in section 4.4. that this nontrivial periodic solution is stable for ᾱ s < 6 5. A plot of the amplitude A as function of ᾱ s is given in figure (4.). This plot confirms the assumption that for ᾱ s = the largest vibration amplitudes occur. In figure (4.) the oscillations for t in (X Y )-plane are given. The 3: internal resonance case As in the previous subsection 4.3. it can easily be shown that the trivial periodic solution is unstable for ᾱ s < 6 5. For the nontrivial periodic solution X(t) = A cos(3θ (t)) and Y (t) = A cos(θ (t)) it follows from the (reduced) linearized map (4.4.3) around this periodic solution that the eigenvalues of this map of (4.4.3) are : b a + 3i ( 8 δ + δ ) and a 3i ( 8 δ + δ ). Since b and a are positive for ᾱ s < 6 5 it follows that the nontrivial periodic solution is stable for ᾱ s < 6 5. In figure (4.3) and in figure (4.4) plots are given of the

83 74 A Weakly Nonlinear Coupled Harmonic Oscillators.5 A α s Figure 4.: Plot of amplitude A as function of ᾱ s. Y A X = Y = A Cos(3 θ (t)) X A Figure 4.: Plot of the stable periodic solution for the :3 internal resonance case. amplitudes A and A of the stable periodic solutions as functions of ᾱ s and the detuning parameter δ = δ 8 + δ. In figure (4.5) plots in the (X Y )-plane are given for the stable periodic solutions of the oscillator for different values of the phase difference θ = θ () θ (). It should be remarked that from figure (4.3) it follows that the largest vibration amplitudes in X-direction do not occur for ᾱ s =. 4.5 Conclusions and remarks In this chapter it has been shown that the perturbation method based on integrating factors can be used efficiently to approximate first integrals for a system of weakly

84 4.5 Conclusions and remarks 75 Amplitude.7.6 A A α s Figure 4.3: Plot of amplitudes A and A as function of ᾱ s for δ =. A δ.5 α s.5 (a) A as function of ᾱ s and δ. (b) A as function of ᾱ s and δ. A δ.5 α s.5 Figure 4.4: Plots of amplitudes as functions of ᾱ s and δ. nonlinear coupled harmonic oscillators. It should be remarked that some of these approximations contain so-called secular terms. In section 4. (and 4.3) of this chapter a justification of the presented perturbation method has been given. It has also been shown how the existence and stability of time-periodic solutions can be deduced from the approximations of the first integrals for a system of weakly nonlinear coupled harmonic oscillators with a :3 or a 3: internal resonance. To obtain these approximations it follows from the computations that it is not necessary to use multiple time-scales. Introducing multiple time-scales can be done (as has been shown in [34]) but a lot of additional computations have to be performed. On the other hand when multiple time-scales are used more accurate (and secular free) approximations of first integrals are obtained on time-scales of order. The ɛ presented perturbation method can easily be applied to other systems of weakly nonlinear coupled harmonic oscillators. In [3] system (4..) has been studied for

85 76 A Weakly Nonlinear Coupled Harmonic Oscillators Y.6 Y.6 Y X X X (a) θ = ᾱ s =. (b) θ = ᾱ s = 5 δ =. (c) θ = 9 ᾱ s = 5 δ =. Figure 4.5: Plots of the stable periodic solution for the 3: internal resonance case. the : : and : internal resonance cases. First order normal form techniques and averaging techniques have been used in [3] to determine the existence and stability of nontrivial periodic solutions. In this chapter we used the recently developed perturbation method based on integrating vectors to study system (4..) with a :3 and a 3: internal resonance. This chapter in fact completes the study of system (4..). It should be remarked that system (4..) is a model that describes galloping of conductor lines (in a windfield) on which ice has accreted. As is well-known galloping is an almost purely vertical oscillation of conductor lines. Our results imply that the system of oscillators will eventually oscillate in an almost purely vertical direction (that is in Y -direction). 4.6 Appendix A - An oscillator with two degrees of freedom in a uniform windfield In [3] a model has been developed to describe the dynamics of an oscillator with two degrees of freedom in a uniform windfield. The following system of a weakly nonlinear coupled harmonic oscillators has been derived Ẍ + ( ω + ɛδ ) ] X = ɛ [ a Ẋ + a Ẏ + a Ẋ a ẊẎ + a Ẏ a 3 Ẏ 3 Ÿ + ( ω + ɛδ ) ] Y = ɛ [ b Ẋ + b Ẏ + b Ẋ b ẊẎ b Ẏ b 3 Ẏ 3 (4.6.) where X = X(t) Y = Y (t) = d dt and where ɛ is a small parameter with < ɛ and where the coefficients a ij and b ij R are given by

86 4.6 Appendix A - An oscillator with two degrees of freedom 77 a = C D > a = C L ᾱ s + C L3 ᾱs 3 a = C D > a = C L ᾱ s + C L3 ᾱs 3 a = C D C L + 3C L3 ᾱs > b = (C L ᾱ s + C L3 ᾱ 3 s) b = (C D + C L + 3 C L3 ᾱ s ) > b = C L ᾱ s + C L3 ᾱ 3 s b = C D C L 3 C L3 ᾱ 3 s > b = 3C L3 ᾱ s C L3 ᾱ3 s C L ᾱ s a 3 = ᾱsc ᾱ L 3C L3 ᾱ s C 3 s L3 b 3 = C D + C L + ( + 6 ᾱ s )C L3 >. The quasi-static drag and lift forces C D (α) and C L (α) acting on a cylinder with ridge can be obtained from wind-tunnel experiments. The coefficients C D C L and C L3 can be derived from these forces (see also section 4.4.3). The angles α α s and ᾱ s are defined in section In figure (4.6) a sketch of the oscillator is presented. The oscillator consists of a cylinder with a small ridge. In figure (4.7) the drag and lift forces acting on the cylinder are given. For more (and complete) details we refer to [6 3 3]. air flow y α s cylinder with ridge x Figure 4.6: The aeroelastic oscillator as viewed from above.

87 78 A Weakly Nonlinear Coupled Harmonic Oscillators axis of symmetry L e L uniform air flow y αs D e D e y e L e x e D x Figure 4.7: Windvelocities and aerodynamic forces acting on the cross section of the cylinder with ridge. 4.7 Appendix B - The :3 internal Resonance In polar coordinates we can rewrite system (4.3.) as follows: dr = ɛ g dt (r r θ θ ) where dθ dt = + ɛ g (r r θ θ ) dr dt = ɛ g 3 (r r θ θ ) dθ dt = + ɛ g 4 (r r θ θ ) (4.7.) g = δ r sin( θ ) a r + a r cos( θ ) + 3 a r cos(θ 3 θ ) 3 a r cos(θ + 3 θ ) a r sin(θ ) 4 a r sin(3 θ ) 3 a r r sin(3 θ ) a r r sin(3 θ + θ ) 3 4 a r r sin( 3 θ + θ ) + 9 a r sin(θ ) 9 4 a r sin(θ + 6 θ ) 9 4 a r sin(θ 6 θ ) 8 8 a 3r 3 cos(θ 3 θ ) a 3r 3 cos(θ + 3 θ ) a 3r 3 cos( 9 θ + θ ) 7 8 a 3r 3 cos(9 θ + θ )

88 4.7 Appendix B - The :3 internal Resonance 79 g = δ cos( θ ) δ a sin( θ ) + 3 a r sin(θ + 3 θ ) r 3 a r sin(θ 3 θ ) + r 4 a r cos(θ ) 4 a r cos(3 θ ) 3 4 a r cos( 3 θ + θ ) a r cos(3 θ + θ ) + 9 a r cos(θ ) 9 r 4 a r cos(θ 6 θ ) 9 r 4 a r cos(θ + 6 θ ) r a r 3 sin(θ + 3 θ ) r 8 a r 3 sin(θ 3 θ ) r a r 3 sin(9 θ + θ ) 7 3 r 8 a r 3 sin( 9 θ + θ ) r g 3 = 6 δ r sin(6 θ ) 6 b r cos(θ 3 θ ) + 6 b r cos(θ + 3 θ ) + b r b r cos(6 θ ) + 6 b r sin(3 θ ) b r sin(3 θ + θ ) + b r sin( 3 θ + θ ) b r r sin(θ ) + 4 b r r sin(θ + 6 θ ) + 4 b r r sin(θ 6 θ ) 9 4 b r sin(3 θ ) b r sin(9 θ ) 7 8 b 3r b 3r 3 cos(6 θ ) 9 8 b 3r 3 cos( θ ) g 4 = 8 δ cos(6 θ ) 8 δ 8 b r sin(θ + 3 θ ) r 8 b r sin(θ 3 θ ) r + 6 b sin(6 θ ) + 8 b r cos(3 θ ) r 36 b r cos( 3 θ + θ ) 36 b r cos(3 θ + θ ) b r cos(θ 6 θ ) + b r cos(θ + 6 θ ) 4 b r cos(3 θ ) + 4 b r cos(9 θ ) 3 4 b 3r sin(6 θ ) b 3r sin( θ ). (4.7.3) The approximations of first integrals in the :3 internal resonance case are F = r + ɛ [ a r t + 4 δ r cos( θ ) + 4 a r sin( θ ) 3 4 a r cos(θ )

89 8 A Weakly Nonlinear Coupled Harmonic Oscillators + a r cos(3 θ ) + a r r cos(3 θ ) 3 a r r cos(3 θ + θ ) 3 4 a r r cos( 3 θ + θ ) 3 4 a r sin(θ 3 θ ) 3 8 a r sin(θ + 3 θ ) 9 a r cos(θ ) a r cos(θ + 6 θ ) 9 a r cos(θ 6 θ ) a 3r 3 sin(θ 3 θ ) a 3r 3 sin(θ + 3 θ ) 7 64 a 3r 3 sin( 9 θ + θ ) 7 8 a 3r 3 sin(9 θ + θ ) ] F = θ + t + ɛ a r + 9 [ δ t δ sin( θ ) + a 4 4 cos( θ ) + a r 4 sin(3 θ ) 3 8 sin(θ ) a r cos(θ + 3 θ ) 3 a r cos(θ 3 θ ) r 4 r a r sin(θ ) + 9 a r sin(θ 6 θ ) 9 a r sin(θ + 6 θ ) r r 8 r + 8 a 3 r 3 cos(θ + 3 θ ) + 8 a 3 r 3 cos(θ 3 θ ) 3 r 6 r 7 a 3 r 3 cos(9 θ + θ ) 7 a 3 r 3 cos( 9 θ + θ ) 8 r 64 r ] + 3a r 4 sin( 3 θ + θ ) + 3a r sin(3 θ + θ ) F 3 = r + ɛ [ b r sin(θ 3 θ ) + 4 b r sin(θ + 3 θ ) 8 b r cos(3 θ ) + 6 b r cos(3 θ + θ ) + b r cos( 3 θ + θ ) + b r r cos(θ ) 8 b r r cos(θ + 6 θ ) + b r r cos(θ 6 θ ) + 36 δ r cos(6 θ ) b r t b 3r 3 t b r sin(6 θ ) b r cos(3 θ ) b r cos(9 θ ) b 3r 3 sin(6 θ ) 3 3 b 3r 3 sin( θ ) ]

90 4.8 Appendix C - The 3: internal resonance case 8 F 4 = θ + t + ɛ [ 8 δ t + b r cos(θ + 3 θ ) b r cos(θ 3 θ ) 7 r 36 r + 54 b r sin(3 θ ) + b r sin( 3 θ + θ ) r 36 r 36 b cos(6 θ ) + 6 b r sin(θ 6 θ ) + 84 b r sin(θ + 6 θ ) b r sin(3 θ ) + 36 b r sin(9 θ ) + 8 b 3r cos(6 θ ) 3 b 3r cos( θ ) 8 δ sin(6 θ ) 8 ] b r sin(3 θ + θ ). (4.7.4) r 4.8 Appendix C - The 3: internal resonance case In polar coordinates system (4.3.8) becomes dr = ɛh dt (r r θ θ ) where dθ dt = + ɛh (r r θ θ ) dr dt = ɛh 3 (r r θ θ ) dθ dt = + ɛh 4 (r r θ θ ) (4.8.5) h = δ r 6 sin(6θ ) a r + a r sin(6θ ) + a r cos(3θ θ ) 6 a r 6 cos(3θ + θ ) a r sin(3θ ) 3 4 a r sin(9θ ) a r r sin(θ ) + 4 a r r sin(θ + 6θ ) 4 a r r sin( θ + 6θ ) + 6 a r sin(3θ ) a r sin(3θ + θ ) a r sin(3θ θ ) 8 a 3r 3 cos(3θ θ ) + 8 a 3r 3 cos(3θ + θ ) + 4 a 3r 3 cos( 3θ + 3θ ) 4 a 3r 3 cos(3θ + 3θ ) h = 8 δ cos(6θ ) 8 δ 6 a sin(6θ ) + 8 a r sin(3θ + θ ) r 8 a r r sin(3θ θ ) + 4 a r cos(3θ ) 4 a r cos(9θ )

91 8 A Weakly Nonlinear Coupled Harmonic Oscillators a r cos( θ + 6θ ) + a r cos(θ + 6θ ) + 8 a cos(3θ ) r 36 a r cos(3θ θ ) r 36 a r cos(3θ + θ ) r 4 a r 3 3 sin(3θ + θ ) + r 4 a r 3 3 sin(3θ θ ) r r a 3 sin(3θ + 3θ ) r 7 a 3 sin(3θ 3θ ) r r 3 r h 3 = δ r sin(θ ) 3 b r cos(3θ θ ) + 3 b r cos(3θ + θ ) + b r r cos(θ ) + 9 b r sin(θ ) 9 4 b r sin(θ + 6θ ) b r sin( θ + 6θ ) 3 b r r sin(3θ ) b r r sin(3θ + θ ) b r r sin(3θ θ ) 3 4 b r sin(θ ) + 4 b r sin(3θ ) 3 8 b 3r 3 + b 3r 3 cos(θ ) 8 b 3r 3 cos(4θ ) h 4 = δ cos(θ ) δ 3 b r sin(3θ + θ ) 3 r b r sin(3θ θ ) r + b sin(θ ) + 9 b r cos(θ ) 9 r 4 b r cos( θ + 6θ ) r 9 4 b r cos(θ + 6θ ) 3 r 4 b r cos(3θ θ ) b r cos(3θ + θ ) 4 b r cos(θ ) + 4 b r cos(3θ ) 4 b 3r sin(θ ) + 8 b 3r sin(4θ ). The approximations of first integrals in the 3: internal resonance case are [ G = r + ɛ 36 δ r cos(6 θ ) + a tr + a r sin(6 θ ) 3 4 a r cos(3 θ ) + a r cos(9 θ ) + a r r cos(θ ) 8 a r r cos(θ + 6 θ ) + a r r cos( θ + 6 θ ) (4.8.6)

92 4.8 Appendix C - The 3: internal resonance case 83 + a r sin(3 θ θ ) 4 a r sin(3 θ + θ ) 8 a r cos(3 θ ) + 6 a r cos(3 θ + θ ) + a r cos(3 θ θ ) 6 a 3r 3 sin(3 θ θ ) + 3 a 3r 3 sin(3 θ + θ ) 4 a 3r 3 t cos( 3 θ + 3 θ ) ] 44 a 3r 3 sin(3 θ + 3 θ ) G = θ + t + ɛ [ 8 δ t + a r sin(3 θ ) 36 a r sin(9 θ ) 7 36 a r cos(3 θ + θ ) + a r cos(3 θ θ ) + a r sin(3 θ ) r 36 r 54 r a r sin(3 θ θ ) a r sin(3 θ + θ ) r 8 r + 96 a 3 r 3 cos(3 θ + θ ) a 3 r 3 cos(3 θ θ ) r 48 r a 3 r 3 cos(3 θ + 3 θ ) + ta 3 r 3 sin( 3 θ + 3 θ ) 43 r 7 r 8 δ sin(6 θ ) + 36 a cos(6 θ ) 6 a r sin( θ + 6 θ ) + ] 84 a r sin(θ + 6 θ ) G 3 = r + ɛ [ 3 4 b r sin(3 θ θ ) b r sin(3 θ + θ ) 9 b r cos(θ ) b r cos(θ + 6 θ ) 9 b r cos( θ + 6 θ ) + b r r cos(3 θ ) 3 b r r cos(3 θ + θ ) 3b r r 4 b tr cos(3 θ θ ) + δ r 4 4 b r sin( θ ) b r cos(θ ) b r cos(3 θ ) b 3r 3 t + 4 b 3r 3 sin( θ ) 3 b 3r 3 sin(4 θ ) ] cos( θ ) G 4 = θ + t + ɛ [ δ t b r cos(3 θ + θ ) r b r cos(3 θ θ ) r

93 84 A Weakly Nonlinear Coupled Harmonic Oscillators + 9 b r sin(θ ) 9 b r sin( θ + 6 θ ) 9 b r sin(θ + 6 θ ) r r 8 r 3 4 b r sin(3 θ θ ) + 3 b r sin(3 θ + θ ) 4 b r sin(θ ) + b r sin(3 θ ) + 8 b 3r cos( θ ) 3 b 3r cos(4 θ ) 4 δ sin( θ ) ] 4 b cos( θ ). (4.8.7)

94 Chapter 5 On Approximations of First Integrals for a Strongly Nonlinear Forced Oscillator Abstract. In this chapter a strongly nonlinear forced oscillator will be studied. It will be shown that the recently developed perturbation method based on integrating factors can be used to approximate first integrals. Not only approximations of first integrals will be given but it will also be shown how in a rather efficient way the existence and stability of time-periodic solutions can be obtained from these approximations. In addition phase portraits Poincaré-return maps and bifurcation diagrams for a set of values of the parameters will be presented. In particular the strongly) nonlinear forced oscillator equation Ẍ + X + λx3 = ɛ (δẋ βẋ3 + γẋ cos(t) will be studied in this chapter. It will be shown that the presented perturbation method not only can be applied to a weakly nonlinear oscillator problem (that is when the parameter λ = O(ɛ)) but also to a strongly nonlinear problem (that is when λ = O()). The model equation as considered in this chapter is related to the phenomenon of galloping of overhead power transmission lines on which ice has accreted. 5. Introduction In this chapter it will be shown how the perturbation method based on integrating vectors can be applied to the following non-autonomous equation Ẍ + du(x) dx = ɛf(x Ẋ t) (5..) This chapter is a revised version of [44] On Approximations of First Integrals for a Strongly Nonlinear Forced Oscillator to be published in Nonlinear Dynamics 3. 85

95 86 Strongly nonlinear forced oscillator where U(X) is the potential energy of the unperturbed (that is ɛ = ) nonlinear oscillator where X = X(t) Ẋ = dx where ɛ is a small parameter satisfying dt < ɛ and where f is a sufficiently smooth function. Many classical perturbation methods have been used to approximate analytically the solution of the weakly nonlinear problem (5..) that is when du(x) in (5..) is linear in X. However dx when du(x) is non-linear in X most of the perturbation methods can not be applied dx to construct approximations of the solutions. In this chapter it will be shown that the perturbation method based on integrating factors can be applied to a class of non-autonomous nonlinear equations as described by (5..). In particular equation (5..) with du(x) = X + dx λx3 and f(x Ẋ t) = δẋ βẋ3 + γẋ cos(ωt) will be studied in detail in this chapter. The existence and stability of time-periodic solutions will be investigated. Bifurcation diagrams will be presented and the dynamics of the oscillator as described by ) Ẍ + X + λx 3 = ɛ (δẋ βẋ3 + γẋ cos(ωt) (5..) will be studied in this chapter. In (5..) it is assumed that the parameter λ is positive that is it is assumed that the oscillator is attached to hard nonlinear springs. The parameters δ > β > γ and ω are assumed to be constants independent of ɛ. The oscillator equation (5..) originates from the following system of ODEs Ÿ + ω Y = ɛ [ a Ẏ + a Ẏ + a Ẋ ] ] Ẍ + ω X + λx3 = ɛ [b Ẋ b Ẏ Ẋ b3ẋ3 (5..3) which is a simple model for the flow-induced vibrations of a cable in a windfield. System (5..3) with λ = or λ = O(ɛ) has been derived by Van der Beek in [3 3]. The coefficients a a a b b and b 3 depend on the quasi-static drag and lift forces acting on a conductor line in a uniform windfield. System (5..3) can be considered to be an oscillator with two degrees of freedom which describes flow-induced vibration of cables in a windfield. The displacement of the cable in vertical direction (that is perpendicular to the windflow) is described by X(t) and the displacement of the cable in horizontal direction (that is in the direction of the windflow) is given by Y (t). For more (and complete) details the reader is referred to [6 3 3]. It is well-known that galloping of conductor lines is an almost purely vertical oscillation of these lines. Based upon the results as obtained in [3 3 43] which confirm this vertical oscillation it is now assumed that the oscillator will oscillate in an almost vertical direction and that Y (t) = A cos(ω t). In this way system (5..3) can be reduced to ) Ẍ + ω X + λx3 = ɛ (b Ẋ b A cos(ω t)ẋ b3ẋ3. (5..4) For small values of λ (that is for λ = O(ɛ)) an interesting internal resonance for system (5..3) is ω : ω = :. This case for instance has been studied in [3 3]. In this chapter it is also assumed that ω : ω = : when λ = O(). After some elementary rescalings in (5..4) equation (5..) is finally obtained with ω =. In

96 5. Introduction 87 particular we will be interested in the existence and stability of (order ) periodic solutions of equation (5..). Many researchers investigated extensively the behavior of the solutions of equations of the type (5..). For instance Nayfeh and Mook [8] Belhaq and Houssni [3] and others investigated the steady-state (periodic) solutions of the weakly nonlinear equation (5..) with du(x) = dx ω ( + h cos(νt))x and f(x Ẋ t) = αẋ βx ξx 3 + γ cos(ωt) using a generalized averaging method or a multiple time-scales perturbation method. For β = h = Lima and Pettini [] studied the control of chaos in a periodically forced oscillator. They showed analytically that a small nonlinear parametric perturbation can suppress chaos. Again by using a multiple time-scales perturbation method Burton and Rahman [7] studied the response of a weakly nonlinear oscillator as described by equation (5..) with du(x) = mx and f(x Ẋ t) = ηẋ g(x) + P cos(ωt) dx where g(x) is an odd nonlinear function and where m is an integer which may be either - or. Roy [9] used an elliptic averaging method to investigate (5..) with du(x) = αx + dx γx3 and f(x Ẋ t) = βẋ + F cos(ωt). Brothers and Haberman [6] also studied (5..) with f(x Ẋ t) = h(x Ẋ) + γ cos(πωt) where h is a purely dissipative perturbation (h is odd in Ẋ) by using averaging and matching techniques. Higher-order averaging techniques based on Lie transforms have been used by Yagasaki and Ichikawa [48] to study weakly nonlinear equations like (5..) with f(x Ẋ t) = δẋ βx αx 3 + γ cos(ωt). Van Horssen [33 34] studied a weakly nonlinear Duffing equation (5..) with f(x Ẋ t) = aẋ bx3 +c cos(t) using the perturbation method based on integrating factors and multiple time-scales. In this chapter it will be shown that for the weakly non-autonomous and weakly nonlinear equation (5..) exactly the same results can be obtained as by applying the classical perturbation techniques (such as averaging multiple time-scales Poincaré-Lindstedt or others). However for the strongly nonlinear equation (5..) with λ = O() most of the classical perturbation methods can not be applied. In this chapter the recently developed perturbation method based on integrating factors (see [33 34]) will be used to construct asymptotic approximations of first integrals for (5..) on long time-scales. In the literature not many analytical results can be found for strongly nonlinear and non-autonomous oscillator equation like (5..). Only recently Yagasaki [47] studied (5..) with λ = and with the perturbation in the right-hand side of (5..) replaced by ( δ + X cos(ωt))ẋ + γ cos(ωt) using an adapted version of Melnikov s method. This chapter is organized as follows. In section 5. of this chapter the construction of approximations of first integrals by using the perturbation method based on integrating factors will be discussed briefly for the general oscillator equation (5..). In section 5.3 approximations of first integrals will be constructed explicitly for the weakly and the strongly nonlinear forced oscillator equation (5..). Using the approximations of the first integrals it will be shown in section 5.4 how the existence and stability of time-periodic solutions for the oscillator equation (5..) can be obtained. The bifurcation(s) of periodic solutions will be studied in detail and a complete set of topological different phase portraits will be presented. Finally in section 5.5 of this chapter some conclusions will be drawn and some remarks will be made.

97 88 Strongly nonlinear forced oscillator 5. Approximations of First Integrals In this section we briefly outline how the perturbation method based on integrating vectors can be applied to approximate first integrals (see also [ ]). We consider the class of non-linear oscillators described by the equation Ẍ + du(x) dx = ɛf(x Ẋ t) (5..) where U(X) is a potential X = X(t) Ẋ = dx ɛ is a small parameter satisfying dt < ɛ and where f is assumed to be sufficiently smooth. We assume that the unperturbed (that is ɛ = ) solutions of (5..) form a family of periodic orbits in the phase-plane (X Ẋ). This family may cover the entire phase plane (X Ẋ) or a bounded region D of the phase plane. Each periodic orbit corresponds to a constant energy level E = + U(X). With each constant energy level E corresponds a phase angle ψ which Ẋ is defined to be ψ = X dr E U(r). (5..) From (5..)-(5..) a transformation (X Ẋ) (E ψ) can then be defined with Ė = ɛẋf = g (E ψ t) [ ψ = + ɛ X dr (E U(r)) 3 ] Ẋf = g (E ψ t). (5..3) Multiplying the first and the second equation in (5..3) with µ (E ψ t) and µ (E ψ t) respectively it follows that the integrating factors µ (E ψ t) and µ (E ψ t) have to satisfy (see also [33 34]) µ = µ ψ E µ t µ t = E (µ g + µ g ) = ψ (µ g + µ g ). (5..4) Let g = ɛg + ɛ g g = ɛg + ɛ g. Expanding µ and µ in formal power series in ɛ that is µ i (E ψ t; ɛ) = µ i (E ψ t) + ɛµ i (E ψ t) +... for i = and substituting g g and the expansions for µ and µ into (5..4) and by taking together terms of equal powers in ɛ we finally obtain the following O(ɛ n )-problems: for n = µ ψ = µ E µ t = µ E (5..5) µ t = µ ψ

98 5. Approximations of First Integrals 89 for n= µ ψ = µ E µ t µ t = E (µ g + µ g + µ ) = ψ (µ g + µ g + µ ) (5..6) and for n µ n = µ n ψ E µ n t µ n t = E (µ n g + µ n g + µ n g + µ n g + µ n ) = ψ (µ n g + µ n g + µ n g + µ n g + µ n ). (5..7) The O(ɛ )-problem (5..5) can readily be solved yielding µ = h (E ψ t) and µ = h (E ψ t) with h = h. The functions h ψ E and h are still arbitrary and will now be chosen as simple as possible. We choose h and h and so (see also [33 4]) µ = µ =. (5..8) It follows from the order ɛ-problem (5..6) that µ and µ have to satisfy µ + µ t ψ µ + µ t ψ = E (g ) = ψ (g ). (5..9) By using the method of characteristics for first order PDEs we then obtain µ = h (E ψ t) t ( (g E ) ) d t µ = h (E ψ t) ( ) (5..) t (g ψ ) d t where h h are arbitrary functions which have to satisfy h ψ t ( ) ψ E (g ) d t = h E t ( ) E ψ (g ) d t. (5..) We choose h and h as simple as possible that is we take h = h =. We then obtain for µ and µ ( µ = t g E d t) ( µ = t g ψ d t). (5..)

99 9 Strongly nonlinear forced oscillator The O(ɛ )-problem (5..7) can be solved yielding ( µ = t (g E + µ g + µ g ) d t) ( ) µ = t (g ψ + µ g + µ g ) d t. (5..3) The O(ɛ n )-problems (5..7) with n > can be solved in a similar way. An approximation F of a first integral F = constant of system (5..3) can now be obtained from (5..8) (5..) and (5..3) yielding (see also [33 34]) [ t ] [ t ] F (E ψ t) = E ɛ g d t ɛ (g + µ g + µ g ) d t.(5..4) How well F approximates a first integral F = constant can be deduced from (see also [33 34]) df dt = [ g + ɛµ g + ɛ µ g + ɛµ g + ɛ µ g ] = ɛ 3 R (E ψ t) (5..5) where g g µ µ µ µ are given by (5..3) (5..) and (5..3) and where the ** indicates that only terms of O(ɛ m ) with m 3 are included. From the existence and uniqueness theorems for ODEs we know that initial value problems for (5..) (with sufficiently smooth potential U(X) and nonlinearity f(x Ẋ t)) are well-posed on a time-scale of order. This implies that also an initial-value problem ɛ for system (5..3) is well-posed on this time-scale. From (5..3) it then follows on this time-scale that if E() is bounded then E(t) is bounded and ψ(t) is bounded by a constant plus t. Since R c + c t on a time scale of order where c ɛ c are constants it follows from (5..5) that and so t F (E(t) ψ(t) t; ɛ) = constant + ɛ 3 R (E(s) ψ(s) s; ɛ)ds F (E(t) ψ(t) t; ɛ) = constant + O(ɛ 3 ) t T < F (E(t) ψ(t) t; ɛ) = constant + O(ɛ) t L ɛ (5..6) where T and L are ɛ-independent constants. Another (functionally independent) approximation of a first integral can be obtained by putting in (5..5) µ = µ = (5..7) instead of (5..8). The O(ɛ)-problem (5..6) can now be solved yielding µ = k (E ψ t) t ( (g E ) ) d t µ = k (E ψ t) ( ) t (g ψ ) d t (5..8)

100 5.3 Approximations of First Integrals for a Nonlinear Forced Oscillator 9 where the functions k and k are arbitrary functions which have to satisfy k ψ t ( ) ψ E (g ) d t = k E t ( ) E ψ (g ) d t. (5..9) We choose these functions as simple as possible that is k = and k =. Then we obtain ( µ = t g E d t) ( µ = t g ψ d t). (5..) The O(ɛ )-problem (5..7) can be solved yielding ( µ = t (g E + µ g + µ g ) d t) ( µ = t (g ψ + µ g + µ g ) d t). (5..) An approximation F of a first integral F = constant of system (5..3) can now be obtained from (5..7) (5..) and (5..) yielding (see also [33 34]) [ t ] [ t ] F (E ψ t) = (ψ t) ɛ g d t ɛ (g + µ g + µ g ) d t.(5..) How well F approximates a first integral F = constant can be deduced from (see also [33 34]) df dt = [ g + ɛµ g + ɛ µ g + ɛµ g + ɛ µ g ] = ɛ 3 R (E ψ t) (5..3) where g g µ µ µ µ are given by (5..3) (5..) and (5..) and where the ** indicates that only terms of O(ɛ m ) with m 3 are included. In the following section we will apply this perturbation method to the oscillator equation (5..). 5.3 Approximations of First Integrals for a Nonlinear Forced Oscillator In this section we will consider the following nonlinear forced oscillator equation Ẍ + du(x) dx where du(x) = ɛf(x Ẋ t) (5.3.) = X + dx λx3 in which λ > is a parameter and where f(x Ẋ t) = δẋ βẋ3 +γẋ cos(t) in which δ > β > and γ are parameters and where ɛ is a small parameter with < ɛ. As explained in the introduction the oscillator equation (5.3.) can be considered to be a simple model describing the vertical displacement of an overhead power transmission line (on which ice has accreted) in

101 9 Strongly nonlinear forced oscillator a windfield. The function X(t) describes the vertical displacement. In this section asymptotic approximations of first integrals for (5.3.) will be constructed explicitly. To give a rather complete analysis of (5.3.) and to understand the bifurcation(s) of the periodic solutions in section 5.4 we will now consider three cases: (i) λ = O(ɛ) (ii) λ = O( ɛ) and (iii) λ = O() The case λ = O(ɛ) Let λ = λɛ with λ >. To study (5.3.) with λ = λɛ in detail we will use straightforward calculations as presented in section 5. to obtain approximations of the first integrals. By introducing the rescalings ɛδ = ɛ X = X β λ = β and γδ = γ it follows that (5.3.) becomes δ λ X + X = ɛ( X β X3 X3 + γ X cos(t)). (5.3.) In the further analysis the tildes will be dropped for convenience. By introducing the transformation (X Ẋ) (E ψ) as defined by E = Ẋ + X ψ = X ( ) dr E r = sin X E (5.3.3) (where E and ψ are the energy and the phase angle of the unperturbed (that is ɛ = ) oscillator respectively) it follows from (5.3.) that Ė = ɛẋg = ξ (E ψ t) = ɛξ (E ψ t) [ ψ = + ɛ X dr (E r ) 3 ] Ẋg = ξ (E ψ t) = + ɛξ (E ψ t) (5.3.4) where g = Ẋ βẋ3 X 3 +γẋ cos(t). From the calculations as presented in section 5. of this chapter it follows that two functionally independent approximations of the first integrals for (5.3.) are given by F (E ψ t) = E ɛ t ξ d t = E ɛ t ( E cos(ψ) 4βE cos(ψ) 4 4E sin(ψ) 3 cos(ψ) +Eγ cos(ψ) cos(t) ) dψ (( = E ɛ E 3 ) ( ) E β ψ + E E β sin(ψ) 8 E β sin(4ψ) + Eγ sin(t) + E + 8 Eγ sin(ψ + t) + ) Eγψ cos(ψ t) (5.3.5)

102 5.3 Approximations of First Integrals for a Nonlinear Forced Oscillator 93 and t F (E ψ t) = (ψ t) ɛ ξ d t = (ψ t) + ɛ t ( E sin(ψ) cos(ψ) E β sin(ψ) cos(ψ) 3 E 4E sin(ψ) 4 + Eγ sin(ψ) cos(ψ) cos(t) ) dψ = (ψ t) + ɛ (( ) Eβ cos(ψ) + 6 Eβ cos(4ψ) + E sin(ψ) 6 E sin(4ψ) 3 4 Eψ + 4 γψ sin(ψ t) ) γ cos(ψ + t). 6 (5.3.6) How well F and F approximate a first integral F = constant can be deduced from df j dt = ɛµ ξ + ɛµ (ξ ) = ɛ R j (E ψ t) (5.3.7) where ξ and ξ are given by (5.3.4). It follows from (5.3.7) that for j = (see also (5..5)-(5..6)) and so t F j (E(t) ψ(t) t; ɛ) = constant + ɛ R j (E(s) ψ(s) s; ɛ)ds (5.3.8) F j (E(t) ψ(t) t; ɛ) = constant + O(ɛ ) t T < F j (E(t) ψ(t) t; ɛ) = constant + O(ɛ) t L ɛ (5.3.9) where T and L are ɛ-independent constants The case λ = O( ɛ) Let λ = ɛ λ with λ >. By introducing the rescalings ɛδ = ɛ X = β λ δ = β and γδ = γ it follows that (5.3.) becomes δ λ X X + X + ɛ X 3 = ɛ( X β X3 + γ X cos(t)). (5.3.) In the further analysis the bars will be dropped for convenience. By introducing the transformation (X Ẋ) (E ψ) as defined by E = Ẋ + X ψ = X ( ) dr E r = sin X E (5.3.)

103 94 Strongly nonlinear forced oscillator (where E and ψ are the energy and the phase angle of the unperturbed (that is ɛ = ) oscillator respectively) it follows from (5.3.) that Ė = ɛẋg = ξ 3(E ψ t) = ɛξ 3 (E ψ t) + ɛξ 3 (E ψ t) ψ = + [ ɛ X dr (E r ) 3 ] Ẋg = ξ 4 (E ψ t) = + ɛξ 4 (E ψ t) + ɛξ 4 (E ψ t) (5.3.) where g = X 3 + ) ɛ (Ẋ β Ẋ 3 + γẋ cos(t). From the calculations as presented in section 5. of this chapter it follows that two functionally independent approximations of the first integrals for system (5.3.) are given by F 3 (E ψ t) = E + t t ɛ ξ 3 d t + ɛ (ξ 3 + µ 3 ξ 3 + µ 4 ξ 4 ) d t = E + t ( ɛ E sin(ψ) ) E sin(4ψ) dψ t ( +ɛ E cos(ψ) E + E β cos(4ψ) + E β cos(ψ) + 3 E β Eγ cos(ψ t) Eγ cos(ψ + t) Eγ cos(t) E3 sin(4ψ) 3 ) 4 E3 sin(ψ) dψ = E + ( ɛ E cos(ψ) + ) 8 E cos(4ψ) +ɛ ((E β ) ( ) 3 E sin(ψ) + E E ψ + 8 E β sin(4ψ) and γψ cos(ψ t) 8 Eγ sin(ψ + t) Eγ sin(t) 3 3 E3 cos(4ψ) + 3 ) 8 E3 cos(ψ) (5.3.3) F 4 (E ψ t) = (ψ t) + t t ɛ ξ 4 d t + ɛ (ξ 4 + µ 3 ξ 3 + µ 4 ξ 4 ) d t = (ψ t) + t ɛ ( 34 E + E cos(ψ) 4 ) E cos(4ψ) dψ t ( +ɛ sin(ψ) 4 βe sin(4ψ) βe sin(ψ) + γ sin(ψ + t) 4 4 γ sin(ψ t) 3 4 E ψ sin(ψ) E ψ sin(4ψ) E cos(4ψ) E 6 3 E cos(ψ) 3 3 E cos(6ψ) + ) 64 E cos(8ψ) dψ = (ψ t) + ( ɛ 3 4 Eψ + E sin(ψ) ) 6 E sin(4ψ)

104 5.3 Approximations of First Integrals for a Nonlinear Forced Oscillator 95 (( +ɛ β + 3 ) 8 E ψ ( + 6 Eβ 3 ) 3 E ψ cos(4ψ) + cos(ψ) ( 3 6 E + 6 ( 3 8 E E 64 E ) sin(ψ) ) sin(4ψ) 6 γ cos(ψ + t) + 4 γψ sin(ψ t) 64 E sin(6ψ) + ) 5 E sin(8ψ). (5.3.4) How well F 3 and F 4 approximate a first integral F = constant can deduced from df j dt = [ ξ j + ɛµ 3 ξ 3 + ɛµ 3 ξ 3 + ɛµ 4 ξ 4 + ɛµ 4 ξ 4 ] = ɛ ɛr j (E ψ t) (5.3.5) where ξ 3 and ξ 4 are given by (5.3.). It follows from (5.3.5) that for j = 3 4 (see also (5..5)-(5..6)) and so F j (E(t) ψ(t) t; ɛ) = constant + ɛ ɛ t R j (E(s) ψ(s) s; ɛ)ds (5.3.6) F j (E(t) ψ(t) t; ɛ) = constant + O(ɛ ɛ) t T < F j (E(t) ψ(t) t; ɛ) = constant + O( ɛ) t L ɛ (5.3.7) where T and L are ɛ-independent constants The case λ = O() In this case the rescalings ɛδ = ˆɛ X = λ ˆX ˆβδλ = β and ˆγδ = γ are introduced and (5.3.) then becomes ˆX + ˆX + ˆX 3 = ˆɛ( ˆX ˆβ ˆX3 + ˆγ ˆX cos(t)). (5.3.8) In the further analysis the heads will be dropped for convenience. By introducing the transformation (X Ẋ) (E ψ) as defined by E = Ẋ + X + 4 X4 ψ = X (5.3.9) dr E r r4 (where E and ψ are the energy and the phase angle of the unperturbed (that is ɛ = ) oscillator) the following system of ODEs is obtained from (5.3.8) Ė = ɛẋg = ξ 5(E ψ t) = ɛξ 5 (E ψ t) [ ψ = + ɛ X dr (E r r4 ) 3 ] Ẋg (5.3.) = ξ 6 (E ψ t) = + ɛξ 6 (E ψ t)

105 96 Strongly nonlinear forced oscillator where g = Ẋ βẋ3 + γẋ cos(t). The solution of the unperturbed (that is ɛ = ) equation (5.3.8) is X = A cn(ϑ k) with ϑ = ω ψ where ψ = t + constant k is a modulus given by k = A and ω ω = + A (see also [8 9 9]). The relationship between the energy E and the amplitude A is given by E = A + 4 A4. The function cn(ϑ k) is a Jacobian elliptic function with argument ϑ and modulus k. From the calculations as presented in section 5. of this chapter it follows that two functionally independent approximations of the first integrals for system (5.3.8) are given by F 5 (E ψ t) = E ɛ t ξ 5 d t and [ t = E ɛ (ω A sn(ϑ k) dn(ϑ k) βω 4 A4 sn(ϑ k)4 dn(ϑ k) 4 ) +γωa sn(ϑ k) dn(ϑ k) cos( ϑ ω) dϑ ] (5.3.) ω ω F 6 (E ψ t) = (ψ t) ɛ t ξ 6 d t = (ψ t) + ɛ [ t P (ϑ k) (ω A sn(ϑ k)dn(ϑ k) βω 3 A3 sn(ϑ k)3 dn(ϑ k) 3) + γω A sn(ϑ k)dn(ϑ k) cos( ϑ ω ω) dϑ where P (ϑ k) = da cn(ϑ k) A de ψsn(ϑ k)dn(ϑ k) dω + A de k sn(ϑ k) and dn(ϑ k) are elliptic functions and where da by da de = A + A 3 dω de = A ω (A + A 3 ) dω de de cn(ϑ k) dk de and dk de dk de = A ( k ) kω (A + A 3 ). ω ] (5.3.) in which are given How well F 5 and F 6 approximate a first integral F = constant can be deduced from df j dt = ɛµ 5 ξ 5 + ɛµ 6 (ξ 6 ) = ɛ R j (E ψ t) (5.3.3) where ξ 5 and ξ 6 are given by (5.3.). It follows from (5.3.3) that for j = 5 6 (see also (5..5)-(5..6)) and so t F j (E(t) ψ(t) t; ɛ) = constant + ɛ R j (E(s) ψ(s) s; ɛ)ds (5.3.4) F j (E(t) ψ(t) t; ɛ) = constant + O(ɛ ) t T < F j (E(t) ψ(t) t; ɛ) = constant + O(ɛ) t L ɛ (5.3.5) where T and L are ɛ-independent constants.

106 5.4 Time-periodic solutions and a bifurcation analysis Time-periodic solutions and a bifurcation analysis In the previous section it has been shown explicitly how asymptotic approximations of first integrals can be obtained. In this section we will show how the existence of non-trivial time-periodic solutions can be determined from the asymptotic approximations of the first integrals. Bifurcation diagrams will be presented and the analytical obtained approximations for the periodic solutions will be compared with numerical results such as obtained by Poincaré map techniques and obtained by numerical integration of the ODEs (phase portraits) The case λ = O(ɛ) The two functionally independent asymptotic approximations (5.3.5) and (5.3.6) for the first integrals of equation (5.3.) can be used to determine the existence and stability of the time-periodic solutions. Moreover from (5.3.5) and (5.3.6) an approximation of a periodic solution can easily be constructed. Let T < be the period of a periodic solution (obviously T should be a multiple of π for γ ). Let G (E ψ t; ɛ) = constant and G (E ψ t; ɛ) = constant be two independent first integrals where G and G are approximated by F and F respectively and where F and F are given by (5.3.5) and (5.3.6) respectively. Let c and c be constants in the two independent first integrals G and G respectively for which a periodic solution exists. Now consider G = c and G = c for t = nt and t = (n )T with n N + : G (E(nT ) ψ(nt ) nt ; ɛ) = c G (E ((n )T ) ψ((n )T ) (n )T ; ɛ) = c G (E(nT ) ψ(nt ) nt ; ɛ) = c G (E ((n )T ) ψ((n )T ) (n )T ; ɛ) = c. (5.4.) Approximating G by F and G by F eliminating c and c from (5.4.) by subtractions we then obtain ( E(nT ) = E ((n )T ) + ɛt E ((n )T ) 3 E ((n )T ) β + γe((n )T ) cos(ψ ((n )T )) ) + O(ɛ t) ψ(nt ) = ψ ((n )T ) T + ɛt ( 3 4 E((n )T ) ) γ sin(ψ ((n )T )) 4 +O(ɛ t) (5.4.) on a time scale of order ɛ. In fact (5.4.) defines a map Q : E Q(E) E n = Q(E n ) which we will use to determine the nontrivial periodic solution(s) of (5.3.).

107 98 Strongly nonlinear forced oscillator By neglecting the O(ɛ t) terms in (5.4.) we can define a new map P : Ẽ P (Ẽ) Ẽ n = P (Ẽn ). It should be remarked that the second equation in the map Q (and in the map P ) will always be considered modulo T. From the well-known theorem of Hartman-Grobman it follows that when the map P has a hyperbolic fixed point then the map Q also has a fixed point which is ɛ-close to the one of the map P. Moreover the fixed point of the map Q has the same stability properties as the corresponding fixed point of the map P. It is also well-known that a fixed point corresponds to a periodic solution of the original ODE that is (5.3.). In this case it follows from (5.4.) with γ that the map P has as nontrivial fixed points (E ψ ) where E = β ± γ (β + ) 4 (5.4.3) 3(β + ) and where ψ is given by γ cos(ψ ) = 3E β and γ sin(ψ ) = 3E. (5.4.4) Since we are interested in nontrivial periodic solutions of (5.3.) (that is E > ) it follows from (5.4.3) that we have to consider the following three cases (a) for γ (β + ) > 4 and < γ < there are two nontrivial solutions for E (b) for γ (β + ) = 4 or γ or γ there is one nontrivial solution for E and (c) for γ (β + ) < 4 and γ there is no nontrivial solution for E. The linearized map of map P around a fixed point of map P is given by ( ) ( 3E β + DP = + ɛt γ cos(ψ ) ) E γ sin(ψ ) 3 γ cos(ψ. (5.4.5) 4 ) By using (5.4.4) it follows from (5.4.5) that the eigenvalues of DP are ( λ = + ɛt 3 βe ± ) 9E. (5.4.6) If the eigenvalues as given by (5.4.6) are not equal to one in modulus then the fixed point (E ψ ) is hyperbolic. The results as given by (5.4.3) and (5.4.4) are exactly the same results as the ones which can be obtained by using the averaging method or the two time-scales perturbation method. The bifurcation diagram in the (β γ)-plane is given in Figure 5.. For E > and ψ < π the following conclusions can be drawn from (5.4.3)-(5.4.6) and from Figure 5.. In region I in Figure 5. we will have one stable fixed point (E ψ ). Crossing the line II a second unstable fixed point is bifurcated. In region III we will have one stable and one unstable fixed point. These two critical points will coincide on the line IV and a saddle node occurs on this line and in region V no fixed points occur. Finally on the line VI that is for γ = we have infinitely many fixed points (E ψ ) with E = 3β and ψ arbitrary. It should be remarked that for γ = equation (5.3.) reduces

108 5.4 Time-periodic solutions and a bifurcation analysis 99 3 I γ V VI II γ = IV 4 β + III V IV II γ = 4 β + III I β Figure 5.: The bifurcation diagram in the (β γ)-plane for the weakly nonlinear forced oscillator equation (5.3.). to well-known autonomous Rayleigh equation. The existence of stable and unstable nontrivial periodic solutions for β = is given in Figure 5. in the (γ E )-plane. In Figure 5.3 phase portraits in the (r ψ)-plane (with E = r ) are given for the first order averaged weakly nonlinear forced oscillator equation (5.3.) with β = and for several values of γ. From Figure 5. and from Figure 5.3 it can readily be seen that we have only one stable periodic solution of (5.3.) for γ > 4 and ψ < π. For γ = 4 a second unstable periodic solution is bifurcated and for 4 < 5 γ < 4 we have two periodic solutions. A saddle node bifurcation occurs for γ = 4 and for < 5 γ < 4 we have no periodic solutions. In Figure 5.4 the 5 Poincaré map technique is used and X(t) and Ẋ(t) are depicted in the (X Ẋ)-plane at times t equal to π 4π 6π 8π.... To compare the analytical results (as given in Figure 5. and Figure 5.) with the numerical results (as given in Figure 5.3 and Figure 5.4) it should be noted that X = r sin(ψ) E = r = (X + Ẋ ). Then it can readily be seen that the analytical results and the numerical results are in good agreement. Finally it should be remarked that an order ɛ approximation of an order π-periodic solution is given by X(t) = E sin(ψ(t)) where E(t) = E + O(ɛ) and ψ(t) = t + ψ + O(ɛ) and where E and ψ are solutions of (5.4.3) and (5.4.4) The case λ = O( ɛ) The two functionally independent asymptotic approximations (5.3.3) and (5.3.4) for the first integrals of equation (5.3.) can be used to approximate the solutions. Moreover from (5.3.3) and (5.3.4) an approximation of a periodic solution (if it exists) can easily be constructed. Let T < be the period of a periodic solution (obviously T should be a multiple of π for γ ). Let G 3 (E ψ t; ɛ) = constant and

109 Strongly nonlinear forced oscillator. Stable periodic solution Unstable periodic solution.8 E γ Figure 5.: The bifurcation diagram in the (γ E )-plane for the weakly nonlinear forced oscillator equation (5.3.) with β =. G 4 (E ψ t; ɛ) = constant be two independent first integrals where G 3 and G 4 are approximated by F 3 and F 4 respectively and where F 3 and F 4 are given by (5.3.3) and (5.3.4) respectively. Let c 3 and c 4 be constants in the two independent first integrals G 3 and G 4 respectively for which a periodic solution exists. Now consider G 3 = c 3 and G 4 = c 4 for t = nt and t = (n )T with n N + : G 3 (E(nT ) ψ(nt ) nt ; ɛ) = c 3 G 3 (E ((n )T ) ψ((n )T ) (n )T ; ɛ) = c 3 G 4 (E(nT ) ψ(nt ) nt ; ɛ) = c 4 G 4 (E ((n )T ) ψ((n )T ) (n )T ; ɛ) = c 4. (5.4.7) Approximating G 3 by F 3 and G 4 by F 4 respectively eliminating c 3 and c 4 from (5.4.7) by subtractions and using the transformation ψ(t) = θ(t) + ɛ 3 te(t) we 4 then obtain ( E(nT ) = E ((n )T ) + ɛt E ((n )T ) 3 E ((n )T ) β + γe((n )T ) cos(θ ((n )T )) ) + O(ɛ ɛt) θ(nt ) = θ ((n )T ) T + ɛt ( 5 64 E((n )T ) ) γ sin(θ ((n )T )) 4 +O(ɛ ɛt) (5.4.8) on a time scale of order ɛ. In fact (5.4.8) defines a map R : E R(E) E n = R(E n ) which we will use to determine the nontrivial periodic solution(s) of

110 5.4 Time-periodic solutions and a bifurcation analysis ψ ψ ψ r r (a) γ =. (b) γ =.5 (c) γ = r ψ ψ ψ r r r (d) γ =.5 (e) γ = (f) γ = ψ ψ ψ r (g) γ = r (h) γ =.5 (i) γ =. Figure 5.3: Phase Portraits in the (r ψ)-plane for the weakly nonlinear forced oscillator equation (5.3.) with β = and for several values of γ. r

111 Strongly nonlinear forced oscillator X X X X (a) γ =. X.5.5 (b) γ = X X (d) γ =.5 X.5.5 (e) γ = X X X (g) γ = X q (f) γ =.5 q X.5 X X.5.5 (c) γ =.5 X.5 X (h) γ = X (i) γ =. Figure 5.4: Poincare -map results for the weakly nonlinear forced oscillator equation (5.3.) in the (X X )-plane for several values of γ with β = and = and with sample-times t equal to π 4π 6π 8π....

112 5.4 Time-periodic solutions and a bifurcation analysis 3 equation (5.3.). By neglecting terms of O(ɛ ɛt) in (5.4.8) we can define a new map S : Ẽ S(Ẽ) Ẽn = S(Ẽn ). It should be remarked that the second equation in the map S (and in the map R) will always be considered modulo T. From the well-known theorem of Hartman-Grobman it follows that when the map S has a hyperbolic fixed point then the map R also has a fixed point which is ɛ-close to the one of the map S. Moreover the fixed point of the map R has the same stability properties as the corresponding fixed point of the map S. In this case it follows from (5.4.8) with γ that the map S has as nontrivial fixed points (E θ ) where E is given by ( (3βE ) + 5 ) 6 E = γ (5.4.9) and where θ is given by γ cos(θ ) = 3E β and γ sin(θ ) = ( ) (5.4.) 5 6 E. The linearized map of map S around a fixed point of map S is given by 3E β + γ cos(θ ) E γ sin(θ ) DP = + ɛt. (5.4.) 5E 3 γ cos(θ ) By using (5.4.9) it follows from (5.4.) that the eigenvalues of DP are ( λ = + ɛt 3 βe ± ) 56 5E 4. 3 (5.4.) Again if the eigenvalues (5.4.) are not equal to one in modules then the fixed point (E θ ) is hyperbolic. The results as given by (5.4.9) and (5.4.) are exactly the same results as the ones which can be obtained by using the second order averaging method or the multiple time-scales method or other perturbation techniques. Using the formulas of Cardano the bifurcation diagram in the (β γ)-plane can be derived from (5.4.9) and is given in Figure 5.5. The regions I-V in Figure 5.5 are as defined in section The existence of stable and unstable nontrivial equilibrium solutions for the nonlinear map (5.4.8) with β = can be determined from Figure 5.6 in the (γ E )-plane. In Figure 5.7 the phase portraits in the (r θ)-plane (with E = r ) are given for the second order averaged nonlinear forced oscillator equation (5.3.) with β = and for several values of γ. It should be remarked that the fixed points as given by (5.4.9) and (5.4.) are not corresponding with the π-periodic solutions of the equation (5.3.) due to the transformation θ(t) = ψ(t) ɛ 3 E(t)t. So the 4 periodic solutions as given in Figure 5.5 and 5.6 for γ are not the π-periodic solutions for the original equation (5.3.). Finally it should be remarked that an approximation of a solution for the nonlinear forced oscillator equation (5.3.) (in a neighborhood of the equilibrium points of the nonlinear map (5.4.8)) is given by X(t) = E sin ( ψ() + t + ɛ 3E 4 t ) + O( ɛ) on a time scale of order ɛ where

113 4 Strongly nonlinear forced oscillator I II III V IV γ VI V IV III II I β Figure 5.5: The bifurcation diagram in the (β γ)-plane for the nonlinear map (5.4.8). E and ψ() = θ() = θ are the solutions of (5.4.9) and (5.4.). We can see from these approximations that the periods of the solutions of (5.3.) (which are O() and not o()) are less than π. This implies that there are no π-periodic solutions which are strict O() (that is are O() but not o()). These results are confirmed in Figure 5.8 in which Poincaré-return map results are given for the nonlinear forced oscillator equation (5.3.) in the (X Ẋ)-plane for several values of γ. It is still possible that equation (5.3.) has small amplitude π-periodic solutions. From the applicational point of view these small amplitude oscillations in vertical direction are not so interesting but from a mathematical point of view these solutions might be of interest to understand the bifurcations that are occurring. To study these small amplitude solutions the following rescaling is usually introduced in (5.3.): X(t) = ɛ α Z(t) with α > yielding ) Z + Z + ɛ +α Z 3 = ɛ (Ż βɛ α Ż 3 + γż cos(t). (5.4.3) The most interesting cases occur for α = and α >. For α = equation (5.4.3) becomes equation (5.3.) with β near zero and this equation has π-periodic solutions for special values of the parameters (see section 5.4.). For α > equation 4 (5.4.3) becomes (up to O(ɛ +α )) ) Z + Z = ɛ (Ż + γ Ż cos(t). (5.4.4) In the Appendix (5.4.4) is studied briefly. From the Poincaré expansion theorem it follows that all solutions of (5.3.) can be expand in X (t)+ ɛx (t)+ɛx (t)+... on a time-scale of order. Obviously Ẍ + X =. So from the Poincaré expansion theorem and the results obtained in this section it follows that equation (5.3.) can only have small amplitude π-periodic solutions as periodic solutions.

114 5.4 Time-periodic solutions and a bifurcation analysis 5 : Stable equilibrium solution : Unstable equilibrium solution E Figure 5.6: The bifurcation diagram in the (γ E )-plane for the nonlinear map (5.4.8) with β = The case λ = O() The two functionally independent asymptotic approximation (5.3.) and (5.3.) for the first integrals of equation (5.3.8) can be used to determine the existence of the time-periodic solutions. Moreover from (5.3.) and (5.3.) an approximation of a periodic solution can easily be constructed. Let T < be the period of a periodic solution (obviously T should be πl with l N + for γ ). Let G 5 (E ψ t; ɛ) = constant and G 6 (E ψ t; ɛ) = constant be two independent first integrals where G 5 and G 6 are approximated by F 5 and F 6 respectively and where F 5 and F 6 are given by (5.3.) and (5.3.) respectively. Let c 5 and c 6 be constants in the two independent first integrals for which a periodic solution exists. Now consider G 5 = c 5 and G 6 = c 6 for t = and t = T. Approximating G 5 by F 5 and G 6 by F 6 (as given by (5.3.) and (5.3.)) eliminating c 5 and c 6 by subtractions we then obtain (using the fact that E() = E(T ) for a periodic solution) ɛ T (Ẋ βẋ4 + γẋ cos(s)) ds = O(ɛ ) ɛ ) T P (ϑ k) (Ẋ β Ẋ 3 + γẋ cos(s) ds = O(ɛ ) γ (5.4.5) where Ẋ = ω A sn(ϑ k)dn(ϑ k). We can rewrite equation (5.4.5) as ɛi(e ψ β γ) = O(ɛ ) ɛj(e ψ β γ) = O(ɛ ). (5.4.6) To have a periodic solution for (5.3.8) we have to find an energy E and a phase angle ψ such that I(E ψ β γ) and J(E ψ β γ) are equal to zero (see also [4 45]).

115 6 Strongly nonlinear forced oscillator θ θ θ r r r (a) γ =. (b) γ =.5 (c) γ = θ θ θ r r r (d) γ =.5 (e) γ = (f) γ = θ θ θ r r r (g) γ =.66 (h) γ =.5 (i) γ =. Figure 5.7: Phase Portraits in the (r θ)-plane for the nonlinear forced oscillator equation (5.3.) with β = and for several values of γ.

116 5.4 Time-periodic solutions and a bifurcation analysis X X X X (a) γ =. X.5.5 (b) γ = X X X.5.5 (e) γ = X X (g) γ =.66.5 X (f) γ =.5.5 X X.5 X (d) γ = (c) γ =.66.5 X.5 X.5 X (h) γ = X (i) γ =. Figure 5.8: Poincare -map results for the nonlinear forced oscillator equation (5.3.) in 5 the (X X )-plane for β = and for several values of γ and = and with sample-times t = π + 4πn where n N.

117 8 Strongly nonlinear forced oscillator To find this energy and phase angle we rewrite I(E ψ β γ) and J(E ψ β γ) in I I βi + γi 3 = (5.4.7) J J βj + γj 3 = where I = T Ẋ ds I = T Ẋ 4 ds I 3 = T J = T P (ϑ k)ẋds J = T P (ϑ k)ẋ3 ds J 3 = T P (ϑ k) (Ẋ cos(s) ) ds. (Ẋ cos(s)) ds (5.4.8) Let D = I 3 J I J 3. It follows from (5.4.7) that for a periodic solution to exist β and γ can be considered to be functions of the energy E and the phase angle ψ that is β = (I D 3J I J 3 ) γ = (I D J I J ) (5.4.9) for D. By using an adaptive recursive Simpson rule the values of the parameters β and γ can be calculated from (5.4.7)-(5.4.9) for which a periodic solution exists. From (5.3.8) it is obvious that the period T should be a multiple of π. The expansion theorem of Poincaré implies that the solution(s) of (5.3.8) can be expanded in X (t) + ɛx (t) + ɛ X (t) +... on a time-scale of order where X satisfies Ẍ + X + X 3 =. Now X (t) is a periodic function with period T (E ) = 4 A dx E X where A > satisfies E A X4 A4 =. In Figure 5.9 T (E ) is plotted. From Figure 5.9 and from the fact that T should be a multiple of π it immediately follows that T should be equal to π or π. For π-periodic solutions it follows from Figure 5.9 that E should be zero and so a π-periodic solution (if it exists) should have small amplitude. To study these small amplitude solutions the following rescaling is introduced in (5.3.8): X(t) = ɛ α Z(t) with α > yielding Z + Z = ɛ α Z 3 + ɛż ɛ+αβż3 + ɛγż cos(t). (5.4.) For π-periodic solutions Z(t) only the case α = and the case α > have to be considered. For α = equation (5.4.) becomes equation (5.3.) with β near zero and this equation has π-periodic solutions for special values of the parameters (see section 5.4.). For α > equation (5.4.4) up to O(ɛα ) is again obtained and this equation has been studied briefly in Appendix. For π-periodic solutions it follows from Figure 5.9 that E should be near and so a π-periodic solution (if it exists) should have an amplitude of (strict) O(). To determine the values of β and γ for which a π-periodic solution exists it should be observed that β = β(e ψ ) and γ = γ(e ψ ) where E = and ψ 4K(k) (in which K(k) is the complete elliptic integral of the first kind). For different values of ψ (with

118 5.5 Conclusions and remarks 9 π T π E Figure 5.9: The period T of the unperturbed equation (5.3.8) (that is (5.3.8) with ɛ = ) as function of the energy E = Ẋ + X + 4 X4. E = ) the integrals in (5.4.8) and (5.4.9) have been calculated by using an adaptive recursive Simpson rule. It should be observed that in (5.4.8) X(t) (that is the solution of the unperturbed equation (5.3.8) with ɛ = ) depends on the initial energy E and on the initial phase angle ψ. For E = β = β(e ψ ) and γ = γ(e ψ ) will give a curve in the (β γ)-plane. This curve has been determined numerically and is given in Figure 5.. From a practical point of view it is obvious that the chance that the parameters β and γ are on this curve is of course zero. For that reason also Poincaré-map results are given in Figure 5. for different values of β and γ. 5.5 Conclusions and remarks In this chapter it has been shown that the perturbation method based on integrating factors can be used efficiently to approximate first integrals for strongly nonlinear forced oscillators. In section 5. (and 5.3) of this chapter a justification of the presented perturbation method has been given. It has also been shown how the existence and stability of time-periodic solutions can be deduced from the approximations of the first integrals for the strongly nonlinear forced oscillators. In this chapter the following three oscillator equations have been studied in detail: ) Ẍ + X = ɛ (Ẋ β Ẋ 3 X 3 + γẋ cos(t) (5.5.) Ẍ + X + ) ɛx 3 = ɛ (Ẋ β Ẋ 3 + γẋ cos(t) and (5.5.) ) Ẍ + X + X 3 = ɛ (Ẋ β Ẋ 3 + γẋ cos(t) (5.5.3)

119 Strongly nonlinear forced oscillator γ β..4 Figure 5.: The curve in the (β γ)-plane for which the strongly nonlinear forced equation (5.3.8) has π-periodic solutions of order. where ɛ is a small parameter with < ɛ and where β > and γ are constants (of order ). In particular the O() behavior of the solutions has been studied. From the applicational point of view this O()-behavior is the most interesting behavior when galloping is studied. For equation (5.5.) it has been shown for what values of the parameters the solutions will tend to a π-periodic solution of order and for what values of the parameters the solutions will tend to a (non-periodic) bounded attractor. The results obtained for (5.5.) are in agreement with the results as obtained in [3 3]. For equation (5.5.) it has been shown that there are no periodic solutions of order. Small amplitude π-periodic solutions however exist for certain values of the parameters. In general the solutions will tend to a bounded non-periodic attractor of order. For equation (5.5.3) it has been shown that there are π-periodic solutions of order for special values of parameters. These π-periodic solutions are however structurally unstable. Also small amplitude π-periodic solutions exist for certain values of parameters. In general the solutions will tend to a bounded non-periodic attractor of order. 5.6 Appendix In section 5.4. and in section the following ODEs have been derived to describe the small amplitude solutions of the oscillator equations ) Z + Z = ɛ (Ż + γ Ż cos(t) ɛ +α Z 3 ɛ +α βż3 and (5.6.) ) Z + Z = ɛ (Ż + γ Ż cos(t) ɛ α Z 3 ɛ +α βż3 (5.6.)

120 5.6 Appendix X X X X X (a) γ =. (b) Zoom in for γ = X (c) γ = X.. X X X (d) Zoom in for γ =.5 (e) γ =..4 (f) γ =.5.4. X X X X X X (g) Zoom in for γ = X (h) γ = X (i) Zoom in for γ =. Figure 5.: Poincare -map results for the nonlinear forced oscillator equation (5.3.8) in and with the (X X )-plane for several values of γ with β = and = + sample-times t = πn with n Z for the figures (a) (c) (e) (f) and (h) and t = πn with n Z+ for the figures (b) (d) (g) and (i).

121 Strongly nonlinear forced oscillator with α > and with α > respectively. In this appendix (5.6.) and (5.6.) will be 4 studied briefly. By introducing the transformation Y (t) = Y (t) cos(t) + Y (t) sin(t) (5.6.3) Ẏ (t) = Y (t) sin(t) + Y (t) cos(t) the first order averaged system of equation (5.6.) or of equation (5.6.) becomes Ẏ = ɛ ( Y γy 4 ) Ẏ = ɛ ( γy 4 + Y ) (5.6.4). For γ not in an o() neighborhood of 4 system (5.6.4) has only as fixed point(s) the trivial fixed point ( ). This fixed point turns out to be unstable. So for γ not in an o() neighborhood of 4 it can be conclude that (5.6.) and (5.6.) do not have nontrivial π-periodic solutions. For γ in an o() neighborhood of 4 second order or higher order averaging has to be applied to (5.6.) and (5.6.). Again it can be shown that (5.6.) and (5.6.) do not have nontrivial π-periodic solutions. The elementary calculations to prove this will be omitted.

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123 4 BIBLIOGRAPHY [3] A. R. Forsyth. Lehrbuch der Differential-Gleichungen. Friedr. Vieweg & Sohn Braunschweig 9. [4] A. R. Forsyth. A treatise on differential equations. Dover Publications Inc. New York 956. [5] J. Guckenheimer and P. Holmes. Nonlinear Oscillations Dynamical Systems and Bifurcation of Vector Fields. Springer-Verlag New York 983. [6] T. I. Haaker and A. H. P. Van der Burgh. Rotational galloping of two coupled oscillators. Meccanica 33: [7] I. D. Iliev and L. M. Perko. Higher order bifurcations of limit cycles. Journal of Differential Equations 54: [8] E. L. Ince and I. N. Sneddon. The solution of ordinary differential equations. Longman Scientific & Technical Harlow Essex 987. [9] W. Kaplan. Ordinary differential equations. Addison-Wesley Massachusetts 958. [] J. Kevorkian and J. D. Cole. Multiple Scale and Singular Perturbation Methods. Springer-Verlag New York 996. [] D. F. Lawden. Elliptic Functions and Applications. Springer-Verlag New York 989. [] R. Lima and M. Pettini. Suppression of chaos by resonance parametric perturbations. Physical Review A 4: [3] S. Lynch. Small amplitude limit cycles of the generalized mixed rayleigh-liénard oscillator. Journal of Sound and Vibration 78(5): [4] J. G. Margallo and J. D. Bejarano. Stability of limit cycles and bifurcations of generalized van der pol oscillator:ẍ + ax bx 3 + ɛ(z 3 + z x + z x 4 )ẋ =. Int. J. Non-Linear Mechanics 5(6): [5] J. G. Margallo and J. D. Bejarano. The limit cycles of the generalized rayleighliénard oscillator. Journal of Sound and Vibration 56(): [6] S. Mitsi S. Natsiavas and I. Tsiafis. Dynamics of nonlinear oscillators under simultaneous internal and external resonances. Nonlinear Dynamics 6: [7] S. Natsiavas. Free vibration of two coupled nonlinear oscillators. Nonlinear Dynamics 6: [8] A. H. Nayfeh and D. T. Mook. Nonlinear Oscillations. John Wiley & Sons New York 985. [9] R. V. Roy. Averaging methods for strongly non-linear oscillators with periodic excitations. Int.J. of Bifurcation and Chaos 9(5):

124 BIBLIOGRAPHY 5 [3] C. G. A. Van der Beek. Normal form and periodic solutions in the theory of non-linear oscillations. existence and asymptotic theory. International journal of non-linear mechanics 4(4): [3] C. G. A. Van der Beek. Analysis of a system of two weakly nonlinear coupled harmonic oscillators arising from the field of wind-induced vibrations. International journal of non-linear mechanics 7(4): [3] W. T. Van Horssen. On integrating factors for ordinary differential equations. Nieuw Archief voor Wiskunde 5: [33] W. T. Van Horssen. A perturbation method based on integrating factors. SIAM journal on Applied Mathematics 59(4): [34] W. T. Van Horssen. A perturbation method based on integrating vectors and multiple scales. SIAM journal on Applied Mathematics 59(4): [35] W. T. Van Horssen. On integrating vectors and multiple scales for singularly perturbed ordinary differential equations. Nonlinear Analysis TMA 46:9 43. [36] W. T. Van Horssen and R. E. Kooij. Bifurcation of limit cycles in a particular class of quadratic systems with two centers. Journal of Differential Equations 4(): [37] W. T. Van Horssen and J. W. Reyn. Bifurcation of limit cycles in a particular class of quadratic systems. Differential and Integral Equations 8(4): [38] F. Verhulst. Nonlinear differential equations and dynamical systems. Springer- Verlag Berlin 996. [39] G. Verros and S. Natsiavas. Self-excited oscillators with asymmetric nonlinearities and one-to-two internal resonance. Nonlinear Dynamics 7: [4] S. B. Waluya and W. T. Van Horssen. On approximations of first integrals for a weakly nonlinear oscillator. In Proceedings of the National Mathematical Conference MIHMI pages ITB Indonesia July 7-. [4] S. B. Waluya and W. T. Van Horssen. On asymptotic approximations of first integrals for a weakly nonlinear oscillator. In Proceedings of the 8th Biennal ASME Conference Symposium on Dynamics and Control of Time-Varying Systems and Structures pages 9 Pittsburgh Pennsylvania USA September 9-. ASME. [4] S. B. Waluya and W. T. Van Horssen. Asymptotic approximations of first integrals for a nonlinear oscillator. Nonlinear Analysis TMA 5(8):

125 6 BIBLIOGRAPHY [43] S. B. Waluya and W. T. Van Horssen. On approximations of first integrals for a system of weakly nonlinear coupled harmonic oscillators. Nonlinear Dynamics 3: [44] S. B. Waluya and W. T. Van Horssen. On approximations of first integrals for a strongly nonlinear forced oscillator. to be published in Nonlinear Dynamics 3. [45] S. B. Waluya and W. T. Van Horssen. On approximations of first integrals for strongly nonlinear oscillators. Nonlinear Dynamics 3: [46] S. Wiggins. Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer-Verlag New York 99. [47] K. Yagasaki. Melnikov s method and codimension-two bifurcations in forced oscillations. Journal of Differential Equations 85: 4. [48] K. Yagasaki and T. Ichikawa. Higher-order averaging for periodically forced weakly nonlinear systems. Int.J. of Bifurcation and Chaos 9(3): [49] S. B. Yuste and J. D. Bejarano. Extension and improvement to the krylovbogoliubov methods using elliptic functions. Int.J. Control 49(4):

126 Summary In this thesis a class of nonlinear oscillator equations is studied. Asymptotic approximations of first integrals for the nonlinear differential equations are constructed by using the recently developed perturbation method based on integrating vectors. The existence and the stability of time-periodic solutions can be determined from these asymptotic approximations of the first integrals. Also asymptotic approximations of the solutions of the oscillator equations can be derived from these asymptotic approximations of the first integrals. Not only autonomous oscillator equations but also nonautonomous equations can be treated. In this thesis it is shown that the presented perturbation method based on integrating vectors can be applied to weakly and strongly nonlinear oscillator equations which are close to integrable equations (that is are integrable in the unperturbed case). In combination with a phase-space analysis and a Poincaré return-map technique the presented perturbation method gives a good insight in the global behavior of the solutions of the oscillator equations. All nonlinear oscillator equations which are studied in this thesis are simple modelequations describing the galloping oscillations of iced overhead power transmission lines in a windfield.

127 8 Summary

128 Samenvatting In dit proefschrift wordt een klasse van niet-lineaire oscillator-vergelijkingen bestudeerd. Asymptotische benaderingen van eerste integralen voor de niet-lineaire differentiaalvergelijkingen worden geconstrueerd met behulp van de recentelijk ontwikkelde storingsmethode welke gebaseerd is op integrerende vectoren. De existentie en de stabiliteit van tijdsperiodieke oplossingen kunnen worden bepaald uit de asymptotische benaderingen van de eerste integralen. Ook kunnen asymptotische benaderingen van de oplossingen van de oscillator-vergelijkingen worden afgeleid uit deze asymptotische benaderingen van de eerste integralen. Niet alleen autonome oscillator-vergelijkingen maar ook niet-autonome vergelijkingen kunnen worden behandeld. In dit proefschrift wordt aangetoond dat de storingsmethode welke gebaseerd is op integrerende vectoren kan worden toegepast op zowel zwak als sterk niet-lineaire oscillator-vergelijkingen welke dicht liggen bij integreerbare vergelijkingen (d.w.z. welke integreerbaar zijn in het ongestoorde geval). In combinatie met een fase-vlak analyse en een Poincaré return-map techniek geeft de gepresenteerde storingsmethode een goed inzicht in het globale gedrag van de oplossingen van de oscillator-vergelijkingen. Alle niet-lineaire oscillator-vergelijkingen die in dit proefschrift bestudeerd worden zijn eenvoudige model vergelijkingen welke de galopperende trillingen van hoogspanningslijnen in een windveld beschrijven.

129 Samenvatting

130 Acknowledgment First of all I would like to thank my supervisor and toegevoegd promotor Dr. ir. Wim T. van Horssen for guiding me in my research and for teaching me mathematics especially nonlinear differential equations and also for his support motivation and patience. Many thanks are also due to Prof. dr. ir. A. J. Hermans for the careful reading of the thesis and the valuable suggestions for improvement. I am indebted to Prof. Dr. F. Verhulst Dr. ir. A. H. P. van der Burgh Dr. ir. T. I. Haaker for their support suggestions and motivation. I would like to thank the Delft University of Technology the Netherlands and the Semarang State University Indonesia (UNNES) for giving me the opportunity to carry out my Ph. D research. This work would not have been possible without the financial support from the Indonesian Government under PGSM project and Centre for International Cooperation and Appropriate Technology (CICAT) TUD the Netherlands. I would like to thank Dr. O. Simbolon Dr. Beny Karyadi Prof. Dr. R. K. Sembiring Drs. H. P. S. Althuis for their assistance to carry out my research. Also thanks to Durk Jellema and Rene Tamboer for their support and help. I am grateful to all members of the Department of Applied Mathematical analysis Faculty of Information Technology and Systems Delft University of Technology especially to Kees Lemmens and Eef Hartman for their support on any computer matter and to Judith C. Ormskerk for secretarial work. Thanks to my colleagues the PGSM group (Differential equations group: Abadi Caswita Darma Gede Happy Hartono Siti and Theo) and also the Statistics group: (Suyono and Ucup) for their time discussion and having fun together. I wish to thank all my friends in the Netherlands for their kind friendship such that I still feel happy although I live far away from my family. Thanks to my friends who give me an opportunity to get something else which is not related with my work but is important for my life.

131 Acknowledgment Finally I am very grateful to my big family especially to my dearest wife Evi and our beloved son Albertus for their love support patience and understanding during my absence from them. All the above can not happen without the power of God so I am very thankful to God that everything becomes possible. Delft Summer 3 Stevanus Budi Waluya

132 Curriculum Vitae Stevanus Budi Waluya was born in Magelang Central Java Indonesia on September He received a Sarjana degree in Mathematics Education in 99. Since 993 he is a staff member of the Department of Mathematics Semarang State University Indonesia (UNNES). In September 994 he started his Master Program at the Bandung Institute of Technology Indonesia. In October 997 he obtained his Master degree in Mathematics. In October 998 he started his Ph.D. study at the Department of Applied Mathematical Analysis of the Delft University of Technology the Netherlands under the supervision of Dr. Ir. Wim T. van Horssen.