Quasiperiodic phenomena in the Van der Pol - Mathieu equation

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1 Quasiperiodic penomena in te Van der Pol - Matieu equation F. Veerman and F. Verulst Department of Matematics, Utrect University P.O. Box , 3508 TA Utrect Te Neterlands April 8, 009 Abstract Te Van der Pol - Matieu equation, combining self-excitation and parametric excitation, is analysed near and at 1 : resonance, using te averaging metod. We analytically prove te existence of stable and unstable periodic solutions near te parametric resonance frequency. Above a certain detuning tresold, quasiperiodic solutions arise wit basic periods of order 1 and order 1/ were is te (small) detuning parameter. 1 Introduction In an early but not widely known monograp, Tondl ([1]) formulated te Van der Pol - Matieu equation to model various engineering problems. Te analysis in [1] employs armonic balance and analogue computer metods. Recently, te Van der Pol - Matieu equation as played an important role in various oter models of dynamical systems wit parametric resonance. Momeni et al. ([]) studied te dynamical beaviour of carged dust grains near parametric resonance, wile Pandey, Rand and Zender ([3]) use te Van der Pol - Matieu equation to model MEMS devices. Te analysis in [] owever is matematically deficient and does not describe all te periodic solutions and bifurcations. Te analysis in [1] and [3] aims at observing a numer of interesting penomena witout proofs. In te present paper, we use averaging and te second Bogoliubov teorem to obtain a more complete picture of te dynamics in te case of small self-excitation and parametric excitation. We locate, approximate and prove te existence of stable and unstable periodic solutions for parametric frequency near te 1: resonance. Interestingly, we find also stable quasiperiodic (multifrequency) solutions on increasing te detuning of te parametric frequency. 1

2 Averaging Following [], we analyse te Van der Pol - Matieu equation d x dt (α β x ) dx dt + ω 0(1 + cos γt)x = 0, (1) were we assume α, β, and ω 0 to be nonnegative. Our goal is to analyse tis equation for small parameter values. We terefore write α = α 0, β = β 0 and = 0 wit α 0, β 0, 0 R + = x R x 0. Furtermore, we consider te parametric excitation frequency to be γ = ω 0 + d. Tis way, we introduce a : 1-resonance wit a small frequency detuning, controlled by te detuning parameter d. We introduce a new timescale τ = (ω 0 + d)t, for wic equation (1) transforms into d x dτ + x = ω 0» (α 0 β 0 x ) dx + (d 0ω0 cos τ)x dτ + O( ) () A number of aspects of tis equation were discussed in [1]. As usual in averaging we introduce slowly varying quantities by x(τ) = a(τ) cos τ + b(τ) sin τ dx dτ (τ) = a(τ) sin τ + b(τ) cos τ (3) wit a and b varying slowly in time. Tis allows us to apply te averaging metod discussed in [] (for a more fundamental treatment see [5]) to obtain da dτ = db dτ = ˆα0 ω 0 a ( 0ω 0 + d) b β 0 (a + b ) a + O( ) ˆα0 ω 0 b ( 0ω 0 d) a β 0 (a + b ) b + O( ) () From tis point on, we will omit te terms of order since treatment up to second order asn t revealed any new penomena; only iger precision is acieved. 3 Equilibrium points Te system of equations () as, next to te trivial equilibrium (a, b) = (0, 0), four nontrivial equilibrium points. If te four equilibria are yperbolic, according to te second Bogoliubov teorem (sometimes called teorem for periodic solutions by averaging ) tey correspond wit periodic solutions wit te same stability caracteristics. Te teorem can be found in [6], capter 6. We follow te formulation in [], teorems , were te proof is based on te implicit function teorem. Teorem Consider te equation and suppose tat: ẋ = f(t, x) + g(t, x, ) x D R n, t 0 (5) a. te vector functions f, g, f x, f x and g x are defined, continuous and bounded by a constant M (independent of ) in [0, ) D, 0 0; b. f(t, x) and g(t, x, ) are T -periodic in t (T independent of );

3 If p is a critical point of te averaged equation wereas ẏ = f 0 (y) (6) f 0 (y) 0 (7) y y=p ten tere exists a T -periodic solution of φ(t, ) of equation (5) wic is close to p suc tat lim 0 φ(t, ) = p. In addition, if te eigenvalues of te critical point y = p of te averaged equation (6) all ave negative real parts, te corresponding periodic solution φ(t, ) of equation (5) is asymptotically stable for sufficiently small. If one of te eigenvalues as positive real part, φ(t, ) is unstable. Notice tat, using te transformation (3) we can bring system () in te desired form (5). Returning to te nontrivial equilibrium points, we divide tese four points into two pairs, wic are labeled symmetric and antisymmetric: v «s a u = ± t r! 0 q 1 α b 0 0 ω 0 ω 0 + d 0 d β ± 0 0ω q A(8) 0 ω 0 d v «a a u = ± t r! 0 q 1 α b ω 0 ω 0 + d 0 d β ± 0 0ω q A(9) 0 ω 0 d We see tat for any nontrivial equilibrium point to exist, te reality condition d 0ω0 (10) must be satisfied. Furtermore, existence of te symmetric pair demands tat r Γ := 0 ω0 d α 0 (11) Tis puts limits on te detuning d of te resonance frequency for te periodic solutions to exist. Stability We determine te stability of te equilibria by computing te eigenvalues at te equilibrium points. For te trivial equilibrium point (a, b) = (0, 0), we find te associated eigenvalues λ 0 ± to be λ 0 ± = [α 0 ± Γ ] (1) ω 0 We see tat if reality condition (10) is not satisfied, te equilibrium point is an unstable focus. If (10) is satisfied but (11) is not, we obtain a saddle point. If bot reality conditions (10) and (11) are satisfied, te equilibrium point is an unstable node. 3

4 For te nontrivial equilibrium points, we observe tat bot points in eac pair exibit te same stability beaviour, because for eac pair, its position and te system of equations () is invariant under te double reflection (a, b) ( a, b). Te eigenvalues for te symmetric (λ s ±) and antisymmetric (λ a ±) pair are λ s ± =» 1 ± 1 α 0 + Γ (13) ω 0 λ a ± =» 1 ± 1 α 0 Γ ω 0 (1) Bot points of te antisymmetric pair are stable nodes, wile bot points of te symmetric pair are saddle points. Te beaviour of te equilibrium points is illustrated in Figure 1. Te relevant bifurcation parameter turns out to be Γ. Looking at our transformation (3), it is clear tat an equilibrium point of te system of equations () corresponds wit a periodic solution of te original equation (). Since we ave found two stable equilibrium points (te antisymmetric pair), we can translate tis into two stable periodic solutions (limit-cycles) of equation (): x(τ) = ± s (α 0 + Γ) β 0 0ω 0 r 0ω 0 + d cos τ r 0ω 0 d sin τ! (15) Te beaviour of tis periodic solution of x is illustrated in Figure. Notice tat te period is equal to π for our rescaled time τ; for te π original time t, te period is = π ω 0 +d ω 0 (1 d ω 0 ) + O( ). As mentioned before, te rigorous existence of tese periodic solutions follows from te second Bogoliubov teorem, see section 3. 5 Quasiperiodic beaviour We look for a stable manifold in te (a, b)-plane wic is invariant under te flow of te system of equations (). We assume tis manifold to be described by a quadric Aa + Bab + Cb = R. Tis assumption turns out to be correct, wit A = α0 d( 0ω 0 d) B = α 0 ω 0 0 C = α0 + d( 0ω 0 + d) R = α 0 β 0 (α0 ( 0 ω 0 d ) (16) Calculating te determinant of te coefficient matrix, we find AC B = (α 0 + d )(α 0 ( 0 ω 0 d )). We conclude tat te quadric is an ellipse wen te reality condition (10) is not met, or wen bot (10) and (11) are satisfied. Tis is equivalent to Γ ir + resp. Γ < α 0. Wen condition (11) is not met (Γ > α 0), te quadric describes a yperbola. Straigtforward calculation sows tat all nontrivial equilibrium points lie on te quadric. Tis means tat several asymptotic solutions in te (a, b)-plane can be identified: Wen bot (10) and (11) are satisfied, system () as four nontrivial equilibria, all of wic lie on te quartic describing an ellipse. Tis situation is depicted in Figure 5. Te ellipse consists of four orbits (a e, b e), for wic lim τ (a e, b e) = (a s, b s ) ± and lim τ (a e, b e) = (a a, b a ) ±.

5 Wen only (10) is satisfied, system () as two nontrivial equilibria (te antisymmetric pair). Eac of tem is located on a different branc of te quartic describing a yperbola. Since bot nontrivial equilibrium points are stable, tey are positive attractors so all orbits converge to one of tese two equilibria. Te symmetry axis between te two brances of te yperbola divides te (a, b)-plane in two stability regions in eac of wic one of te equilibrium points is te only attractor. In te caption of Figure 1 we identify te corresponding subcritical pitcfork and saddle-node bifurcations. In addition, if te reality condition (10) is not met (no nontrivial equilibrium points exist), te quadric (wic is an ellipse in tis case) contains a periodic orbit. Tis follows from te fact tat, for tese parameter values, te origin is an unstable focus, wile for a, b 1 (outside te ellipse) te direction field points inwards. Applying te Poincaré-Bendixson teorem to te averaged system () yields te existence of a periodic orbit on te ellipse. Tis situation is depicted in Figure 6. Te periodic orbit corresponds wit a torus in te original system () as follows from [5], appendix C. Te period of tis orbit can be found if we write te system of equations () in polar coordinates. Coosing a(τ) = r(τ) cos θ(τ) and b(τ) = r(τ) sin θ(τ), we obtain dr dτ dθ dτ = = ω 0 ω 0» α 0 0ω0» d 0ω0 cos θ «sin θ r β0 r3 Substituting y(τ) = tan θ(τ), we can solve equation (18), yielding s d + 0ω 0 «d 0ω 0 tan θ(τ) = tan Γ (τ τ 0) + φ 0 ω 0 wit tan φ 0 = q d 0 ω 0 (17) (18) (19) r d+ 0 ω 0 tan θ(τ d 0 ω 0 0). Notice tat, since (10) is not satisfied, Γ =. From (19) we infer tat te frequency ω of tis orbit is ω = ω 0 Γ, so te period of te orbit is π ω = πω 0. Γ Tis means tat x(τ) exibits quasiperiodic beaviour. We can distinguis two frequencies: te first one equal to unity for our time scale τ, te second one of order, equal to ω 0 Γ. Tis beaviour is illustrated in Figure 3; a Poincaré section is depicted in Figure. Notice tat tis beaviour only occurs wen no nontrivial equilibrium points in te (a, b)- plane exist. Tis is equivalent to te situation tat te reality condition (10) is not satisfied, so tat te detuning d is above te tresold 0ω 0 For our original time scale t, te new period becomes d ω 0 ) + O(). πω 0 (ω 0 +d) Γ = π. (1 Γ 5

6 6 Conclusion We studied equation (1) for small (order ) values of α, β and. As in [] we considered te main resonance frequency γ = ω 0 + d, wic means tat we are perturbing around te 1: resonance. As new features wit respect to [], we found stable periodic and stable quasi-periodic solutions. Tese more complicated modulations, quasiperiodic solutions, arise if te detuning crosses a certain tresold. References [1] A. Tondl. On te interaction between self-excited and parametric vibrations. Number 5 in Monograps and Memoranda, National Researc Institute for Macine Design, Bĕcovice. Prague, [] M. Momeni, I. Kourakis, M. Moslei-Fard, and P.K. Sukla. A Van der Pol-Matieu equation for te dynamics of dust grain carge in dusty plasmas. J. Pys. A: Mat. Teor., 0:F73 F81, 007. [3] M Pandey, R.H. Rand, and A. Zender. Frequency locking in a forced Matieu - van der Pol - Duffing system. Nonlinear Dynamics, 5:3 1, 008. [] F. Verulst. Nonlinear Differential Equations and Dynamical Systems. Springer, 006. [5] J.A. Sanders, F. Verulst, and J. Murdock. Averaging Metods in Nonlinear Dynamical Systems. Springer, New York, 007. [6] N.N. Bogoliubov and J.A. Mitropolskii. Asymptotic metods in te teory of non-linear oscillations. Gordon and Breac, New York, Acknowledgement Following te comments of one of te referees, we ave added more information and clarification to our exposition. 6

7 Figure 1: Bifurcation analysis of te equilibrium points of te system of equations (). Te points of te symmetric pair are indicated by s ± and tose of te antisymmetric pair by a ±. Te relevant bifurcation parameter is Γ = 0 ω 0 d. We see tat if Γ < α 0, four nontrivial equilibrium points exist wile te origin is an unstable node. For Γ > α 0, only two nontrivial equilibrium points exist; te origin as turned into a saddle point. We can terefore identify a subcritical pitcfork bifurcation at Γ = α 0 and two simultaneous saddle-node bifurcations at Γ = 0. Figure : Te periodic beaviour of x(τ), as described by equation (15). Te function is plotted for α 0 = 1, β 0 = 1, 0 =, ω 0 = and d = 1. Te sign in (15) is cosen to be positive. 7

8 Figure 3: Te quasiperiodic beaviour of x(τ). Te function is plotted for α 0 = 1, β 0 = 1, 0 =, ω 0 = and d = 1, wile = 0.1. Since d > 0ω0, a and b exibit periodic beaviour. Tis period of order 1 is indicated in te figure. Te initial frequency of order 1 is also visible. Figure : Te Poincare section (also known as te time=π -map or stroboscopic map) based on te original system (1), of te quasiperiodic solution of x(τ) in pase space (x(τ), x (τ)). Te first 13 iterations are sown. In tis plot, ω 0 = 1. 8

9 Figure 5: Te pase plane of te system () as been drawn for α 0 = 1, β 0 = 1, ω 0 = 1, 0 = 1 and d = 0. In tis case, Γ < α 0 so all four nontrivial equilibrium points exist. Te quadric, wic is an ellipse for tese parameter values, is visible. Te four nontrivial equilibrium points lie on te ellipse. For tese parameter values te origin is an unstable node, bot points of te symmetric pair are saddle points and bot points of te antisymmetric pair are stable nodes. 9

10 Figure 6: Te pase plane of te system () as been drawn for α 0 = 1, β 0 = 1, ω 0 = 1, 0 = and d = 1. In tis case, d > 0ω0 so no nontrivial equilibrium points exist.. Te quadric, wic is an ellipse for tese parameter values, is clearly visible. All drawn solutions converge to te periodic elliptical orbit. 10

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