2:1:1 Resonance in the Quasi-Periodic Mathieu Equation
|
|
- Horatio Singleton
- 5 years ago
- Views:
Transcription
1 Nonlinear Dynamics 005) 40: c pringer 005 :: Resonance in the Quasi-Periodic Mathieu Equation RICHARD RAND and TINA MORRION Department of Theoretical & Applied Mechanics, Cornell University, Ithaca, NY 4853, U..A.; Author for correspondence rhr@cornell.edu; fax: ) Received: July 004; accepted: 7 October 004) Abstract. We present a small ɛ perturbation analysis of the quasi-periodic Mathieu equation ẍ + δ + ɛ cos t + ɛ cos ωt)x = 0 in the neighborhood of the point δ = 0.5 and ω = 0.5. We use multiple scales including terms of Oɛ ) with three time scales. We obtain an asymptotic expansion for an associated instability region. Comparison with numerical integration shows good agreement for ɛ = 0.. Then we use the algebraic form of the perturbation solution to approximate scaling factors which are conjectured to determine the size of instability regions as we go from one resonance to another in the δ ω parameter plane. Key words: parametric excitation, resonance, quasi-periodic Mathieu equation. Introduction The following quasi-periodic Mathieu equation, ẍ + δ + ɛ cos t + ɛ cos ωt)x = 0 ) has been the topic of a number of recent research papers [ 7]. In particular, the stability of Equation ) has been investigated. For given parameters ɛ, δ, ω), Equation ) is said to be stable if all solutions are bounded, and unstable if an unbounded solution exists.) A stability chart for ɛ = 0. isshown in Figure. This chart was obtained in [] by numerical integration of Equation ). A striking feature of this complicated figure is that various details appear to be repeated at different length scales. For example, the shape of the instability regions around the point δ = 0.5 and ω = appear to be similar to those near the point δ = 0.5 and ω = 0.5, except that the latter regions are smaller in scale. The purpose of this work is to investigate this similarity, and to obtain a scaling law which relates these two regions. Our approach will be based on approximate solutions obtained by perturbation methods. In the case of the instability regions near the point δ = 0.5 and ω =, a perturbation analysis has been performed in [5], giving approximate closed form expressions for the transition curves separating regions of stability from regions of instability. For example, the following approximate expression was derived for the largest of the local instability regions near δ = 0.5 and ω =, see Figure : ω = + ɛ ± + δ )) + Oɛ ) )
2 96 R. Rand and T. Morrison Figure. tability diagram of Equation ) from []. White regions are unstable, black regions are stable. Figure. Transition curves near δ = 0.5 and ω = based on Equation ) from [5]. Compare with Figure. where δ = 4 + δ ɛ and = 4δ 3) In the present work we use a different perturbation method to obtain a comparable approximate expression for the largest of the local instability regions near δ = 0.5 and ω = 0.5. By comparing the two expressions we are able to determine scaling factors which relate their relative sizes.
3 Quasi-Periodic Mathieu Equation 97 The title of this paper is explained by noting that Equation ) may be viewed as an oscillator with natural frequency δ which is parametrically forced with frequencies and ω. Near the point δ = 0.5 and ω = 0.5, the three frequencies are in the ratio of ::.. Perturbation Method We use the method of multiple time scales on Equation ), with the goal of obtaining an approximate expression for transition curves in the neighborhood of δ = 0.5 and ω = 0.5. To accomplish this, we found it necessary to go to Oɛ ) and to use three time scales: ξ = t, η= ɛt, and ζ = ɛ t 4) As usual in this method [8] the dependent variable x becomes a function of ξ, η, and ζ,givinganew expression for the second derivative of x in Equation ): d x dt = x ξ + ɛ x ξ η + x ɛ η + x ɛ ξ ζ + Oɛ3 ) 5) We expand x, ω, and δ as a power series of ɛ: x = x 0 + ɛ x + ɛ x + ω = + ɛω + ɛ ω + δ = 4 + ɛδ + ɛ δ + 6) and substitute these series into Equations ), 5) and collect terms in ɛ. Using subscripts to represent partial differentiation, we obtain: ɛ 0 : x 0,ξξ + 4 x 0 = 0 7) ɛ : x,ξξ + 4 x = x 0,ξη δ x 0 x 0, cos ξ ) ξ x 0 cos + ω η + ω ζ ɛ : x,ξξ + 4 x = x 0,ξζ x 0,ηη x,ξη δ x 0 δ x ) ξ x cos ξ x cos + ω η + ω ζ 8) 9) We write the solution to Equation 7) in the form x 0 ξ,η,ζ) = Aη, ζ) cos ξ + Bη, ζ) sin ξ 0) where A and B are as yet undetermined slowly varying coefficients. ubstituting Equation 0) into Equation 8) and removing secular terms gives the following equations on A and B: A η = δ ) B ) B η = δ + ) A
4 98 R. Rand and T. Morrison Equations ) have the solution: Aη, ζ) = A ζ ) cos η + A ζ ) sin η ) and a similar expression for Bη, ζ). Here = 4δ and A and A are as yet undetermined slowly varying coefficients. Having removed secular terms from Equation 8), we may solve for x, which may be written in the abbreviated form: x ξ,η,ζ) = Cη, ζ) cos ξ + Dη, ζ) sin ξ + periodic terms 3) where the first two terms on the RH of Equation 3) are the complementary solution of Equation 8) involving as yet undetermined slowly varying coefficients Cη, ζ) and Dη, ζ). The periodic terms represent a particular solution of Equation 8). Although the expressions for these are too long give here, we note that they consist of sinusoidal terms with arguments 3 ξ ± η, ω η + ω ζ ± η, and ξ + ω η + ω ζ ± η 4) Next, we substitute the expressions for x 0 and x, Equations 0) and 3), and the expressions for A and B, Equation ), into the x equation, Equation 9), and eliminate secular terms. This gives equations on C and D which may be written in the following form: C η = δ ) D + periodic terms 5) D η = δ + ) C + periodic terms Although the expressions for the periodic terms are too long to give here, we note that they consist of sinusoidal terms with arguments η, ω ± ) η + ω ζ, and ω ± ) η ω ζ 6) Note that Equations 5) on the arbitrary coefficients C, D of x are similar in form to the Equations ) on the arbitrary coefficients A,B of x 0,except that the C-D eqs. are nonhomogeneous. Thus our next step is to remove secular terms from Equations 5). ince = 4δ, we see that the periodic terms in Equations 5) which have argument η are resonant. For general values of ω, these are the only resonant terms. However, if ω = /, then some of the other periodic terms in Equations 5) will be resonant as well, because in that case ω )η = η. We first consider the case in which ω does not equal /. Eliminating secular terms in Equations 5) turns out to give the following equations on the slow flow coefficients A and A which appeared in the expressions for A and B, Equation ): A ζ = pa 7) A ζ = pa
5 Quasi-Periodic Mathieu Equation 99 where p = δ 4δ 4δ 7) 8) ince all solutions to Equations 7) are periodic and bounded, no instability is possible. We next consider the resonant case in which ω = /. Elimination of secular terms in Equations 5) gives: A ζ = pa + q A sin ω ζ + A cos ω ζ ) A ζ = pa + q A sin ω ζ + A cos ω ζ ) 9) where p is given by Equation 8), and where q is given by the following equation: q = + δ 0) As we show in the next section, Equations 9) exhibit unbounded solutions. Parameter values for which unbounded solutions occur in 9) correspond to regions of instability in the stability diagram. 3. Analysis of the low Flow In order to investigate the stability of the slow flow system 9), we write it in the form: u = pv + q u sin t + v cos t) v = pu + qv sin t + u cos t) ) where A, A,ω and ζ have been replaced respectively by u, v, and t for convenience. We begin by transforming ) to polar coordinates, u = r cos θ, v = r sin θ: ṙ = qr sin t θ) θ = p + q cos t θ) ) Next we replace θ by φ = t θ: ṙ = qr sin φ φ = p q cos φ 3) Writing Equations 3) in first order form, dr dφ = qr sin φ p q cos φ 4)
6 00 R. Rand and T. Morrison we are easily able to obtain the integral: r = const p q cos φ 5) Equation 5) represents a curve in the u v plane. If the denominator of the RH vanishes for some value of φ, then the curve extends to infinity and the corresponding motion is unstable. This case corresponds to the existence of an equilibrium point in the φ equation the second of Equations 3)). In the contrary case in which the denominator of the RH of Equation 5) does not vanish, the motion remains bounded stable) and the φ equation has no equilibria. Thus the condition for instability is that the following equation has a real solution: cos φ = p q 6) The transition case between stable and unstable is given by the condition: p q =± = p ± q 7) ubstituting Equations 8),0) and using = ω, Equation 7) becomes: ω = δ 4δ 4δ 7) ± + δ ) 8) where = 4δ. In addition, for resonance we required that ω = 9) Thus we obtain the following expression for transition curves near δ = 0.5 and ω = 0.5: ω = + ɛ + 4δ 4δ ɛ δ 7) ± + δ ) ) + 30) where δ = 4 + ɛδ + ɛ δ + 3) Equation 30) gives a value of ω for a given value of δ, the latter defined by δ, δ and ɛ.however, for a given value of ɛ,any value of δ close to 0.5 can be achieved in the form /4 + ɛδ, that is by choosing δ = 0. o in our numerical evaluation of Equations 30) and 3), we take δ = 0. ee Figure 3 where the transition curves 30) are displayed for ɛ = 0. along with the results of previous work, Equation ), already displayed in Figure cf. Figure.
7 Quasi-Periodic Mathieu Equation 0 Figure 3. Transition curves near δ = 0.5 and ω = 0.5 based on Equation 30). Also shown for comparison are the transition curves near δ = 0.5 and ω = based on Equation ) from [5]. Compare with Figures and. 4. Discussion We now have asymptotic approximations for corresponding instability regions at two points in the δ ω parameter plane, and we wish to compare them. Based on an expansion about the point δ = 0.5 and ω =, we have Equation ) from [5]. And from the work presented in this paper, we have Equation 30), valid in the neighborhood of δ = 0.5 and ω = 0.5. We repeat these expansions here for the convenience of the reader, using the subscript A for Equation ), and the subscript B for Equation 30): ω A = + ɛ ± + δ )) + Oɛ ) 3) ω B = + ɛ + ɛ δ 4δ 7 ) ± + δ )) + Oɛ 3 ) 33) Note that although these expansions are truncated at different orders of ɛ, both are the lowest order approximations which respectively yield two transition curves as represented by the ± sign), and which therefore allow the thickness of the associated instability region to be computed. We compare these two expressions in two ways: ) the centerline of the region, and ) the thickness of the region. The respective centerlines are given by the following approximations: ω A = + ɛ + Oɛ ) 34) ω B = + ɛ + Oɛ ) 35)
8 0 R. Rand and T. Morrison From these we may conjecture that the scaling of the centerline in the ω direction goes like ω 0, being the ω value of the point of expansion. That is, in the case of Equation 34), ω 0 =, while for Equation 35), ω 0 = /, and we observe that the two curves have comparable shape, but that they are stretched in the ω direction in proportion to their values of ω 0. Moving on to the question of the thickness of the two instability regions, these are obtained by subtracting the expressions for the upper and lower transition curves, and are given by the following approximations: thickness A = ɛ + δ ) thickness B = ɛ + δ + Oɛ ) 36) ) + Oɛ 3 ) 37) We see that once again the two expressions have the same general form, but that the smaller region is a factor of ɛ thinner than the larger region. This leads us to the conjecture that this is due to the difference in the order of the resonance. For example, this would lead to the guess that the comparable instability region associated with the point δ = 0.5 and ω = /3 which would correspond to a : 3 : resonance) would have a similar equation for its thickness, but with an ɛ 3 in the leading term. 5. Conclusions We have presented a small ɛ perturbation analysis of the quasi-periodic Mathieu equation ) in the neighborhood of the point δ = 0.5 and ω = 0.5. We used multiple scales and found that we needed to go to Oɛ ) and use three time scales in order to obtain a minimal representation of an instability region. Comparison with numerical integration of Equation ) showed good agreement for ɛ = 0.. We used the perturbation approximation to estimate the scaling of instability regions as we go from one resonance to another in the δ ω parameter plane. This is an interesting use of perturbation approximations. A comparable result could not be easily achieved by purely numerical methods. For example, inspection of Figure would lead us to conclude that the instability region near δ = 0.5 and ω = ismuch thicker than the comparable region near δ = 0.5 and ω = 0.5. However, it would be difficult to conclude using only numerical results that the ratio of thicknesses was approximately ɛ, as we showed in this work. References. Zounes, R.. and Rand, R. H., Transition curves in the quasi-periodic Mathieu equation, IAM Journal of Applied Mathematics 58, 998, Rand, R., Zounes, R., and Hastings, R., A quasi-periodic Mathieu equation, Chap. 9, in Nonlinear Dynamics: The Richard Rand 50th Anniversary Volume, A.Guran ed.), World cientific Publications, 997, pp Mason,. and Rand, R., On the torus flow Y = A + B CO Y + C CO X and its relation to the quasi-periodic Mathieu equation, in Proceedings of the 999 AME Design Engineering Technical Conferences, Las Vegas, NV, eptember 5, 999, Paper no. DETC99/VIB-805 CD-ROM). 4. Zounes, R.. and Rand, R. H., Global behavior of a nonlinear quasi-periodic Mathieu equation, Nonlinear Dynamics 7, 00, Rand, R., Guennoun, K., and Belhaq, M., :: Resonance in the quasi-periodic Mathieu equation, Nonlinear Dynamics 3, 003,
9 Quasi-Periodic Mathieu Equation Guennoun, K., Houssni, M., and Belhaq, M., Quasi-periodic solutions and stability for a weakly damped nonlinear quasiperiodic Mathieu equation, Nonlinear Dynamics 7, 00, Belhaq, M., Guennoun, K., and Houssni, M., Asymptotic solutions for a damped non-linear quasi-periodic Mathieu equation, International Journal of Non-Linear Mechanics 37, 00, Rand, R. H., Lecture Notes on Nonlinear Vibrations version 45), Published on-line by The Internet-First University Press, Ithaca, NY, 004,
2:2:1 Resonance in the Quasiperiodic Mathieu Equation
Nonlinear Dynamics 31: 367 374, 003. 003 Kluwer Academic Publishers. Printed in the Netherlands. ::1 Resonance in the Quasiperiodic Mathieu Equation RICHARD RAND Department of Theoretical and Applied Mechanics,
More informationTwo models for the parametric forcing of a nonlinear oscillator
Nonlinear Dyn (007) 50:147 160 DOI 10.1007/s11071-006-9148-3 ORIGINAL ARTICLE Two models for the parametric forcing of a nonlinear oscillator Nazha Abouhazim Mohamed Belhaq Richard H. Rand Received: 3
More informationDYNAMICS OF THREE COUPLED VAN DER POL OSCILLATORS WITH APPLICATION TO CIRCADIAN RHYTHMS
Proceedings of IDETC/CIE 2005 ASME 2005 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference September 24-28, 2005, Long Beach, California USA DETC2005-84017
More informationDynamics of Two Coupled van der Pol Oscillators with Delay Coupling Revisited
Dynamics of Two Coupled van der Pol Oscillators with Delay Coupling Revisited arxiv:1705.03100v1 [math.ds] 8 May 017 Mark Gluzman Center for Applied Mathematics Cornell University and Richard Rand Dept.
More informationChapter 14 Three Ways of Treating a Linear Delay Differential Equation
Chapter 14 Three Ways of Treating a Linear Delay Differential Equation Si Mohamed Sah and Richard H. Rand Abstract This work concerns the occurrence of Hopf bifurcations in delay differential equations
More informationCoexistence phenomenon in autoparametric excitation of two
International Journal of Non-Linear Mechanics 40 (005) 60 70 wwwelseviercom/locate/nlm Coexistence phenomenon in autoparametric excitation of two degree of freedom systems Geoffrey Recktenwald, Richard
More informationDynamicsofTwoCoupledVanderPolOscillatorswithDelayCouplingRevisited
Global Journal of Science Frontier Research: F Mathematics and Decision Sciences Volume 7 Issue 5 Version.0 Year 07 Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals
More informationThree ways of treating a linear delay differential equation
Proceedings of the 5th International Conference on Nonlinear Dynamics ND-KhPI2016 September 27-30, 2016, Kharkov, Ukraine Three ways of treating a linear delay differential equation Si Mohamed Sah 1 *,
More informationA Model of Evolutionary Dynamics with Quasiperiodic Forcing
paper presented at Society for Experimental Mechanics (SEM) IMAC XXXIII Conference on Structural Dynamics February 2-5, 205, Orlando FL A Model of Evolutionary Dynamics with Quasiperiodic Forcing Elizabeth
More informationTHREE PROBLEMS IN NONLINEAR DYNAMICS WITH 2:1 PARAMETRIC EXCITATION
THREE PROBLEMS IN NONLINEAR DYNAMICS WITH 2:1 PARAMETRIC EXCITATION A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the
More information2:1 Resonance in the delayed nonlinear Mathieu equation
Nonlinear Dyn 007) 50:31 35 DOI 10.1007/s11071-006-916-5 ORIGINAL ARTICLE :1 Resonance in the delayed nonlinear Mathieu equation Tina M. Morrison Richard H. Rand Received: 9 March 006 / Accepted: 19 September
More informationCoupled Parametrically Driven Modes in Synchrotron Dynamics
paper presented at Society for Experimental Mechanics SEM) IMAC XXXIII Conference on Structural Dynamics February -5, 015, Orlando FL Coupled Parametrically Driven Modes in Synchrotron Dynamics Alexander
More informationFrequency locking in a forced Mathieu van der Pol Duffing system
Nonlinear Dyn (008) 54:3 1 DOI 10.1007/s11071-007-938-x ORIGINAL ARTICLE requency locking in a forced Mathieu van der Pol Duffing system Manoj Pandey Richard H. Rand Alan T. Zehnder Received: 4 August
More informationAvailable online at ScienceDirect. Procedia IUTAM 19 (2016 ) IUTAM Symposium Analytical Methods in Nonlinear Dynamics
Available online at www.sciencedirect.com ScienceDirect Procedia IUTAM 19 (2016 ) 11 18 IUTAM Symposium Analytical Methods in Nonlinear Dynamics A model of evolutionary dynamics with quasiperiodic forcing
More informationLinear and Nonlinear Oscillators (Lecture 2)
Linear and Nonlinear Oscillators (Lecture 2) January 25, 2016 7/441 Lecture outline A simple model of a linear oscillator lies in the foundation of many physical phenomena in accelerator dynamics. A typical
More informationDETC DELAY DIFFERENTIAL EQUATIONS IN THE DYNAMICS OF GENE COPYING
Proceedings of the ASME 2007 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2007 September 4-7, 2007, Las Vegas, Nevada, USA DETC2007-3424
More informationAnswers to Problem Set Number MIT (Fall 2005).
Answers to Problem Set Number 6. 18.305 MIT (Fall 2005). D. Margetis and R. Rosales (MIT, Math. Dept., Cambridge, MA 02139). December 12, 2005. Course TA: Nikos Savva, MIT, Dept. of Mathematics, Cambridge,
More informationJournal of Applied Nonlinear Dynamics
Journal of Applied Nonlinear Dynamics 5(3) (016) 337 348 Journal of Applied Nonlinear Dynamics https://lhscientificpublishing.com/journals/jand-default.aspx Delay-Coupled Mathieu Equations in Synchrotron
More information8 Example 1: The van der Pol oscillator (Strogatz Chapter 7)
8 Example 1: The van der Pol oscillator (Strogatz Chapter 7) So far we have seen some different possibilities of what can happen in two-dimensional systems (local and global attractors and bifurcations)
More informationMultiple scale methods
Multiple scale methods G. Pedersen MEK 3100/4100, Spring 2006 March 13, 2006 1 Background Many physical problems involve more than one temporal or spatial scale. One important example is the boundary layer
More informationMIT Weakly Nonlinear Things: Oscillators.
18.385 MIT Weakly Nonlinear Things: Oscillators. Department of Mathematics Massachusetts Institute of Technology Cambridge, Massachusetts MA 02139 Abstract When nonlinearities are small there are various
More informationJournal of Applied Nonlinear Dynamics
Journal of Applied Nonlinear Dynamics 4(2) (2015) 131 140 Journal of Applied Nonlinear Dynamics https://lhscientificpublishing.com/journals/jand-default.aspx A Model of Evolutionary Dynamics with Quasiperiodic
More informationDynamics of a mass-spring-pendulum system with vastly different frequencies
Dynamics of a mass-spring-pendulum system with vastly different frequencies Hiba Sheheitli, hs497@cornell.edu Richard H. Rand, rhr2@cornell.edu Cornell University, Ithaca, NY, USA Abstract. We investigate
More informationJournal of Applied Nonlinear Dynamics
Journal of Applied Nonlinear Dynamics 5(4) (6) 47 484 Journal of Applied Nonlinear Dynamics https://lhscientificpublishing.com/journals/jand-default.aspx Delay erms in the Slow Flow Si Mohamed Sah, Richard
More informationP321(b), Assignement 1
P31(b), Assignement 1 1 Exercise 3.1 (Fetter and Walecka) a) The problem is that of a point mass rotating along a circle of radius a, rotating with a constant angular velocity Ω. Generally, 3 coordinates
More informationHigher Order Averaging : periodic solutions, linear systems and an application
Higher Order Averaging : periodic solutions, linear systems and an application Hartono and A.H.P. van der Burgh Faculty of Information Technology and Systems, Department of Applied Mathematical Analysis,
More informationM.ARCIERO, G.LADAS AND S.W.SCHULTZ
Georgian Mathematical Journal 1(1994), No. 3, 229-233 SOME OPEN PROBLEMS ABOUT THE SOLUTIONS OF THE DELAY DIFFERENCE EQUATION x n+1 = A/x 2 n + 1/x p n k M.ARCIERO, G.LADAS AND S.W.SCHULTZ Abstract. We
More informationDYNAMICS OF THREE COUPLED VAN DER POL OSCILLATORS WITH APPLICATION TO CIRCADIAN RHYTHMS
DYNAMICS OF THREE COUPLED VAN DER POL OSCILLATORS WITH APPLICATION TO CIRCADIAN RHYTHMS A Thesis Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements
More informationSolving the Van Der Pol nonlinear differential equation using first order approximation perturbation method
Solving the Van Der Pol nonlinear differential equation using first order approximation perturbation method Nasser M. Abbasi 2009 compiled on Monday January 01, 201 at 07:16 PM The Van Der Pol differential
More informationMethod of Averaging for Differential Equations on an Infinite Interval
Method of Averaging for Differential Equations on an Infinite Interval Theory and Applications SUB Gottingen 7 222 045 71X ;, ' Vladimir Burd Yaroslavl State University Yaroslavl, Russia 2 ' 08A14338 Contents
More informationDynamics of four coupled phase-only oscillators
Communications in Nonlinear Science and Numerical Simulation 13 (2008) 501 507 www.elsevier.com/locate/cnsns Short communication Dynamics of four coupled phase-only oscillators Richard Rand *, Jeffrey
More informationCDS 101 Precourse Phase Plane Analysis and Stability
CDS 101 Precourse Phase Plane Analysis and Stability Melvin Leok Control and Dynamical Systems California Institute of Technology Pasadena, CA, 26 September, 2002. mleok@cds.caltech.edu http://www.cds.caltech.edu/
More informationCDS 101/110a: Lecture 2.1 Dynamic Behavior
CDS 11/11a: Lecture.1 Dynamic Behavior Richard M. Murray 6 October 8 Goals: Learn to use phase portraits to visualize behavior of dynamical systems Understand different types of stability for an equilibrium
More informationDETC NONLINEAR DYNAMICS OF A SYSTEM OF COUPLED OSCILLATORS WITH ESSENTIAL STIFFNESS NONLINEARITIES
Proceedings of DETC 2003 2003 ASME Design Engineering Technical Conferences September 2-6, 2003, Chicago, Illinois, USA DETC2003-4446 NONLINEAR DYNAMICS OF A SYSTEM OF COUPLED OSCILLATORS WITH ESSENTIAL
More informationCDS 101/110a: Lecture 2.1 Dynamic Behavior
CDS 11/11a: Lecture 2.1 Dynamic Behavior Richard M. Murray 6 October 28 Goals: Learn to use phase portraits to visualize behavior of dynamical systems Understand different types of stability for an equilibrium
More informationSolutions for B8b (Nonlinear Systems) Fake Past Exam (TT 10)
Solutions for B8b (Nonlinear Systems) Fake Past Exam (TT 10) Mason A. Porter 15/05/2010 1 Question 1 i. (6 points) Define a saddle-node bifurcation and show that the first order system dx dt = r x e x
More informationLectures on Periodic Orbits
Lectures on Periodic Orbits 11 February 2009 Most of the contents of these notes can found in any typical text on dynamical systems, most notably Strogatz [1994], Perko [2001] and Verhulst [1996]. Complete
More informationAdditive resonances of a controlled van der Pol-Duffing oscillator
Additive resonances of a controlled van der Pol-Duffing oscillator This paper has been published in Journal of Sound and Vibration vol. 5 issue - 8 pp.-. J.C. Ji N. Zhang Faculty of Engineering University
More informationChapter 2 PARAMETRIC OSCILLATOR
CHAPTER- Chapter PARAMETRIC OSCILLATOR.1 Introduction A simple pendulum consists of a mass m suspended from a string of length L which is fixed at a pivot P. When simple pendulum is displaced to an initial
More informationOrdinary differential equations
Class 7 Today s topics The nonhomogeneous equation Resonance u + pu + qu = g(t). The nonhomogeneous second order linear equation This is the nonhomogeneous second order linear equation u + pu + qu = g(t).
More informationSuppression of the primary resonance vibrations of a forced nonlinear system using a dynamic vibration absorber
Suppression of the primary resonance vibrations of a forced nonlinear system using a dynamic vibration absorber J.C. Ji, N. Zhang Faculty of Engineering, University of Technology, Sydney PO Box, Broadway,
More informationNORMAL MODES, WAVE MOTION AND THE WAVE EQUATION. Professor G.G.Ross. Oxford University Hilary Term 2009
NORMAL MODES, WAVE MOTION AND THE WAVE EQUATION Professor G.G.Ross Oxford University Hilary Term 009 This course of twelve lectures covers material for the paper CP4: Differential Equations, Waves and
More informationα Cubic nonlinearity coefficient. ISSN: x DOI: : /JOEMS
Journal of the Egyptian Mathematical Society Volume (6) - Issue (1) - 018 ISSN: 1110-65x DOI: : 10.1608/JOEMS.018.9468 ENHANCING PD-CONTROLLER EFFICIENCY VIA TIME- DELAYS TO SUPPRESS NONLINEAR SYSTEM OSCILLATIONS
More informationProblem Set 1 October 30, 2017
1. e π can be calculated from e x = x n. But that s not a good approximation method. The n! reason is that π is not small compared to 1. If you want O(0.1) accuracy, then the last term you need to include
More informationTable of contents. d 2 y dx 2, As the equation is linear, these quantities can only be involved in the following manner:
M ath 0 1 E S 1 W inter 0 1 0 Last Updated: January, 01 0 Solving Second Order Linear ODEs Disclaimer: This lecture note tries to provide an alternative approach to the material in Sections 4. 4. 7 and
More informationRelative Stability of Mathieu Equation in Third Zone
Relative tability of Mathieu Equation in Third Zone N. Mahmoudian, Ph.D. tudent, Nina.Mahmoudian@ndsu.nodak.edu Tel: 701.231.8303, Fax: 701.231-8913 G. Nakhaie Jazar, Assistant Professor, Reza.N.Jazar@ndsu.nodak.edu
More information7 Pendulum. Part II: More complicated situations
MATH 35, by T. Lakoba, University of Vermont 60 7 Pendulum. Part II: More complicated situations In this Lecture, we will pursue two main goals. First, we will take a glimpse at a method of Classical Mechanics
More informationForced Mechanical Vibrations
Forced Mechanical Vibrations Today we use methods for solving nonhomogeneous second order linear differential equations to study the behavior of mechanical systems.. Forcing: Transient and Steady State
More informationPart III. Interaction with Single Electrons - Plane Wave Orbits
Part III - Orbits 52 / 115 3 Motion of an Electron in an Electromagnetic 53 / 115 Single electron motion in EM plane wave Electron momentum in electromagnetic wave with fields E and B given by Lorentz
More informationAssignment 6. Using the result for the Lagrangian for a double pendulum in Problem 1.22, we get
Assignment 6 Goldstein 6.4 Obtain the normal modes of vibration for the double pendulum shown in Figure.4, assuming equal lengths, but not equal masses. Show that when the lower mass is small compared
More informationAn introduction to Birkhoff normal form
An introduction to Birkhoff normal form Dario Bambusi Dipartimento di Matematica, Universitá di Milano via Saldini 50, 0133 Milano (Italy) 19.11.14 1 Introduction The aim of this note is to present an
More informationNon-linear Dynamics in Queueing Theory: Determining Size of Oscillations in Queues with Delayed Information
Non-linear Dynamics in Queueing Theory: Determining Size of Oscillations in Queues with Delayed Information Sophia Novitzky Center for Applied Mathematics Cornell University 657 Rhodes Hall, Ithaca, NY
More informationAn analysis of queues with delayed information and time-varying arrival rates
Nonlinear Dyn https://doi.org/10.1007/s11071-017-401-0 ORIGINAL PAPER An analysis of queues with delayed information and time-varying arrival rates Jamol Pender Richard H. Rand Elizabeth Wesson Received:
More informationNONLINEAR NORMAL MODES OF COUPLED SELF-EXCITED OSCILLATORS
NONLINEAR NORMAL MODES OF COUPLED SELF-EXCITED OSCILLATORS Jerzy Warminski 1 1 Department of Applied Mechanics, Lublin University of Technology, Lublin, Poland, j.warminski@pollub.pl Abstract: The main
More informationLecture 6 Asymptotic Structure for Four-Step Premixed Stoichiometric Methane Flames
Lecture 6 Asymptotic Structure for Four-Step Premixed Stoichiometric Methane Flames 6.-1 Previous lecture: Asymptotic description of premixed flames based on an assumed one-step reaction. basic understanding
More informationCoupled Oscillators Monday, 29 October 2012 normal mode Figure 1:
Coupled Oscillators Monday, 29 October 2012 In which we count degrees of freedom and find the normal modes of a mess o masses and springs, which is a lovely model of a solid. Physics 111 θ 1 l m k θ 2
More informationFigure 5.16 Compound pendulum: (a) At rest in equilibrium, (b) General position with coordinate θ, Freebody
Lecture 27. THE COMPOUND PENDULUM Figure 5.16 Compound pendulum: (a) At rest in equilibrium, (b) General position with coordinate θ, Freebody diagram The term compound is used to distinguish the present
More informationDifference Resonances in a controlled van der Pol-Duffing oscillator involving time. delay
Difference Resonances in a controlled van der Pol-Duffing oscillator involving time delay This paper was published in the journal Chaos, Solitions & Fractals, vol.4, no., pp.975-98, Oct 9 J.C. Ji, N. Zhang,
More informationReduction in number of dofs
Reduction in number of dofs Reduction in the number of dof to represent a structure reduces the size of matrices and, hence, computational cost. Because a subset of the original dof represent the whole
More informationTopic 5: The Difference Equation
Topic 5: The Difference Equation Yulei Luo Economics, HKU October 30, 2017 Luo, Y. (Economics, HKU) ME October 30, 2017 1 / 42 Discrete-time, Differences, and Difference Equations When time is taken to
More informationResonances of a Forced Mathieu Equation with Reference to Wind Turbine Blades
Resonances of a Forced Mathieu Equation with Reference to Wind Turbine Blades Venkatanarayanan Ramakrishnan and Brian F Feeny Dynamics Systems Laboratory: Vibration Research Department of Mechanical Engineering
More information1. < 0: the eigenvalues are real and have opposite signs; the fixed point is a saddle point
Solving a Linear System τ = trace(a) = a + d = λ 1 + λ 2 λ 1,2 = τ± = det(a) = ad bc = λ 1 λ 2 Classification of Fixed Points τ 2 4 1. < 0: the eigenvalues are real and have opposite signs; the fixed point
More information1 Lyapunov theory of stability
M.Kawski, APM 581 Diff Equns Intro to Lyapunov theory. November 15, 29 1 1 Lyapunov theory of stability Introduction. Lyapunov s second (or direct) method provides tools for studying (asymptotic) stability
More informationIn which we count degrees of freedom and find the normal modes of a mess o masses and springs, which is a lovely model of a solid.
Coupled Oscillators Wednesday, 26 October 2011 In which we count degrees of freedom and find the normal modes of a mess o masses and springs, which is a lovely model of a solid. Physics 111 θ 1 l k θ 2
More informationEN Nonlinear Control and Planning in Robotics Lecture 3: Stability February 4, 2015
EN530.678 Nonlinear Control and Planning in Robotics Lecture 3: Stability February 4, 2015 Prof: Marin Kobilarov 0.1 Model prerequisites Consider ẋ = f(t, x). We will make the following basic assumptions
More informationA FIELD METHOD FOR SOLVING THE EQUATIONS OF MOTION OF EXCITED SYSTEMS UDC : (045) Ivana Kovačić
FACTA UNIVERSITATIS Series: Mechanics, Automatic Control and Robotics Vol.3, N o,, pp. 53-58 A FIELD METHOD FOR SOLVING THE EQUATIONS OF MOTION OF EXCITED SYSTEMS UDC 534.:57.98(45) Ivana Kovačić Faculty
More informationTHE STABILITY OF PARAMETRICALLY EXCITED SYSTEMS: COEXISTENCE AND TRIGONOMETRIFICATION
THE STABILITY OF PARAMETRICALLY EXCITED SYSTEMS: COEXISTENCE AND TRIGONOMETRIFICATION A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the
More informationAn Equation with a Time-periodic Damping Coefficient: stability diagram and an application
An Equation with a Time-periodic Damping Coefficient: stability diagram and an application Hartono and A.H.P. van der Burgh Department of Technical Mathematics and Informatics, Delft University of Technology,
More informationA plane autonomous system is a pair of simultaneous first-order differential equations,
Chapter 11 Phase-Plane Techniques 11.1 Plane Autonomous Systems A plane autonomous system is a pair of simultaneous first-order differential equations, ẋ = f(x, y), ẏ = g(x, y). This system has an equilibrium
More informationCitation Acta Mechanica Sinica/Lixue Xuebao, 2009, v. 25 n. 5, p The original publication is available at
Title A hyperbolic Lindstedt-poincaré method for homoclinic motion of a kind of strongly nonlinear autonomous oscillators Author(s) Chen, YY; Chen, SH; Sze, KY Citation Acta Mechanica Sinica/Lixue Xuebao,
More information1. Comparison of stability analysis to previous work
. Comparison of stability analysis to previous work The stability problem (6.4) can be understood in the context of previous work. Benjamin (957) and Yih (963) have studied the stability of fluid flowing
More informationSeminar 6: COUPLED HARMONIC OSCILLATORS
Seminar 6: COUPLED HARMONIC OSCILLATORS 1. Lagrangian Equations of Motion Let consider a system consisting of two harmonic oscillators that are coupled together. As a model, we will use two particles attached
More informationLong-wave Instability in Anisotropic Double-Diffusion
Long-wave Instability in Anisotropic Double-Diffusion Jean-Luc Thiffeault Institute for Fusion Studies and Department of Physics University of Texas at Austin and Neil J. Balmforth Department of Theoretical
More informationAn Application of Perturbation Methods in Evolutionary Ecology
Dynamics at the Horsetooth Volume 2A, 2010. Focused Issue: Asymptotics and Perturbations An Application of Perturbation Methods in Evolutionary Ecology Department of Mathematics Colorado State University
More informationMath 575-Lecture 26. KdV equation. Derivation of KdV
Math 575-Lecture 26 KdV equation We look at the KdV equations and the so-called integrable systems. The KdV equation can be written as u t + 3 2 uu x + 1 6 u xxx = 0. The constants 3/2 and 1/6 are not
More informationOSCILLATORS WITH DELAY COUPLING
Proceedings of DETC'99 1999 ASME Design Engineering Technical Conferences September 1-15, 1999, Las Vegas, Nevada, USA D E T C 9 9 / V IB -8 3 1 8 BIFURCATIONS IN THE DYNAMICS OF TWO COUPLED VAN DER POL
More informationWork sheet / Things to know. Chapter 3
MATH 251 Work sheet / Things to know 1. Second order linear differential equation Standard form: Chapter 3 What makes it homogeneous? We will, for the most part, work with equations with constant coefficients
More informationChapter 1. Introduction to Nonlinear Space Plasma Physics
Chapter 1. Introduction to Nonlinear Space Plasma Physics The goal of this course, Nonlinear Space Plasma Physics, is to explore the formation, evolution, propagation, and characteristics of the large
More information1 Pushing your Friend on a Swing
Massachusetts Institute of Technology MITES 017 Physics III Lecture 05: Driven Oscillations In these notes, we derive the properties of both an undamped and damped harmonic oscillator under the influence
More informationEN Nonlinear Control and Planning in Robotics Lecture 10: Lyapunov Redesign and Robust Backstepping April 6, 2015
EN530.678 Nonlinear Control and Planning in Robotics Lecture 10: Lyapunov Redesign and Robust Backstepping April 6, 2015 Prof: Marin Kobilarov 1 Uncertainty and Lyapunov Redesign Consider the system [1]
More informationProf. Krstic Nonlinear Systems MAE281A Homework set 1 Linearization & phase portrait
Prof. Krstic Nonlinear Systems MAE28A Homework set Linearization & phase portrait. For each of the following systems, find all equilibrium points and determine the type of each isolated equilibrium. Use
More informationHUYGENS PRINCIPLE FOR THE DIRICHLET BOUNDARY VALUE PROBLEM FOR THE WAVE EQUATION
Scientiae Mathematicae Japonicae Online, e-24, 419 426 419 HUYGENS PRINCIPLE FOR THE DIRICHLET BOUNDARY VALUE PROBLEM FOR THE WAVE EQUATION KOICHI YURA Received June 16, 24 Abstract. We prove Huygens principle
More informationWe denote the space of distributions on Ω by D ( Ω) 2.
Sep. 1 0, 008 Distributions Distributions are generalized functions. Some familiarity with the theory of distributions helps understanding of various function spaces which play important roles in the study
More informationIn-Plane and Out-of-Plane Dynamic Responses of Elastic Cables under External and Parametric Excitations
Applied Mathematics 5, 5(6): -4 DOI:.59/j.am.556. In-Plane and Out-of-Plane Dynamic Responses of Elastic Cables under External and Parametric Excitations Usama H. Hegazy Department of Mathematics, Faculty
More informationExercises Lecture 15
AM1 Mathematical Analysis 1 Oct. 011 Feb. 01 Date: January 7 Exercises Lecture 15 Harmonic Oscillators In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium
More informationLecture 2 Supplementary Notes: Derivation of the Phase Equation
Lecture 2 Supplementary Notes: Derivation of the Phase Equation Michael Cross, 25 Derivation from Amplitude Equation Near threshold the phase reduces to the phase of the complex amplitude, and the phase
More informationPower system modelling under the phasor approximation
ELEC0047 - Power system dynamics, control and stability Thierry Van Cutsem t.vancutsem@ulg.ac.be www.montefiore.ulg.ac.be/~vct October 2018 1 / 16 Electromagnetic transient vs. phasor-mode simulations
More informationMA 201: Partial Differential Equations D Alembert s Solution Lecture - 7 MA 201 (2016), PDE 1 / 20
MA 201: Partial Differential Equations D Alembert s Solution Lecture - 7 MA 201 (2016), PDE 1 / 20 MA 201 (2016), PDE 2 / 20 Vibrating string and the wave equation Consider a stretched string of length
More informationNonlinear Dynamic Systems Homework 1
Nonlinear Dynamic Systems Homework 1 1. A particle of mass m is constrained to travel along the path shown in Figure 1, which is described by the following function yx = 5x + 1x 4, 1 where x is defined
More informationQuasiperiodic phenomena in the Van der Pol - Mathieu equation
Quasiperiodic penomena in te Van der Pol - Matieu equation F. Veerman and F. Verulst Department of Matematics, Utrect University P.O. Box 80.010, 3508 TA Utrect Te Neterlands April 8, 009 Abstract Te Van
More informationASYMPTOTIC THEORY FOR WEAKLY NON-LINEAR WAVE EQUATIONS IN SEMI-INFINITE DOMAINS
Electronic Journal of Differential Equations, Vol. 004(004), No. 07, pp. 8. ISSN: 07-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) ASYMPTOTIC
More informationMCE/EEC 647/747: Robot Dynamics and Control. Lecture 8: Basic Lyapunov Stability Theory
MCE/EEC 647/747: Robot Dynamics and Control Lecture 8: Basic Lyapunov Stability Theory Reading: SHV Appendix Mechanical Engineering Hanz Richter, PhD MCE503 p.1/17 Stability in the sense of Lyapunov A
More informationNon-linear dynamics of a system of coupled oscillators with essential stiness non-linearities
Available online at www.sciencedirect.com International Journal of Non-Linear Mechanics 39 (2004) 1079 1091 Non-linear dynamics of a system of coupled oscillators with essential stiness non-linearities
More informationLyapunov Stability Theory
Lyapunov Stability Theory Peter Al Hokayem and Eduardo Gallestey March 16, 2015 1 Introduction In this lecture we consider the stability of equilibrium points of autonomous nonlinear systems, both in continuous
More informationLinearization problem. The simplest example
Linear Systems Lecture 3 1 problem Consider a non-linear time-invariant system of the form ( ẋ(t f x(t u(t y(t g ( x(t u(t (1 such that x R n u R m y R p and Slide 1 A: f(xu f(xu g(xu and g(xu exist and
More information2 Lyapunov Stability. x(0) x 0 < δ x(t) x 0 < ɛ
1 2 Lyapunov Stability Whereas I/O stability is concerned with the effect of inputs on outputs, Lyapunov stability deals with unforced systems: ẋ = f(x, t) (1) where x R n, t R +, and f : R n R + R n.
More informationTHREE PROBLEMS OF RESONANCE IN COUPLED OR DRIVEN OSCILLATOR SYSTEMS
THREE PROBLEMS OF RESONANCE IN COUPLED OR DRIVEN OSCILLATOR SYSTEMS A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the
More informationWaves and characteristics: Overview 5-1
Waves and characteristics: Overview 5-1 Chapter 5: Waves and characteristics Overview Physics and accounting: use example of sound waves to illustrate method of linearization and counting of variables
More informationM A : Ordinary Differential Equations
M A 2 0 5 1: Ordinary Differential Equations Essential Class Notes & Graphics D 19 * 2018-2019 Sections D07 D11 & D14 1 1. INTRODUCTION CLASS 1 ODE: Course s Overarching Functions An introduction to the
More informationKey words. quasi-periodic, Floquet theory, Hill s equation, Mathieu equation, perturbations, stability
SIAM J. APPL. MATH. c 1998 Society for Industrial and Applied Mathematics Vol. 58, No. 4, pp. 1094 1115, August 1998 004 TRANSITION CURVES FOR THE QUASI-PERIODIC MATHIEU EQUATION RANDOLPH S. ZOUNES AND
More information