Energy Exchanges and Localization in Forced Nonlinear Oscillators and Strongly Degenerate Systems

1. Energy Exchanges and Localizationin Forced Nonlinear Oscillators and Strongly Degenerate Systems YuliStarosvetskyFaculty of Mechanical Engineering,Technion Israel Institute of Technology,Technion City, Haifa 2nd Conference on Localized Excitations in Nonlinear Complex Systems Sevilla, July 10, 2012

2. Prof. L.I. Manevitch N.N. Semenov Institute of Chemical Physics, Russian Academy of Sciences, Moscow, 119991, Russia Mr. Tzahi Ben-Meir Faculty of Mechanical Engineering,Technion Israel Institute of Technology, Haifa, 32000, Israel Collaborators:

3. Forced Duffing Oscillator L. I. Manevitch, A. S. Kovaleva, and D. S. Shepelev, PhysicaD 240, 1 (2011). Complexvariables: We introduce a multi-scale expansion: Slow Evolution:

4. Forced Duffing Oscillator Polarrepresentation:  = aei  fora > 0 andthetransformedequationsofmotion: where Δ =  - (s1-π/2) Intheabsenceofdissipation (γ = 0), thereexistsanintegralofmotion Stationarystates: Limitingphasetrajectories: (LPT) H=0

5. a 1 0.8 0.6 0.4 0.2 t 1 10 20 30 40 50 60 70 Forced Duffing Oscillator Nonlinearoscillatorwithoutdissipationbeforeandafterthe firsttopologicaltransition α = 0.093Phase trajectoriesfors=0.4, F=0.13, α = 0.094 α = 0.093α = 0.094

6. Forced Duffing Oscillator Second Topological Transition … Phase trajectoriesfors=0.4, F=0.13, α = 0.187

7. Forced Duffing Oscillator Concept of Limiting Phase Trajectories – Two Distinct Types of LPTs Transition from one type of LPT to another one (Manevitch et. al.) Thus: Quasi-Linear Oscillations Non-Linear Oscillations

8. Forced Duffing Oscillator Non-smoothbasis: a) b) Nonsmoothbasis functions a)  (1); b) e (1) Timedependencesoftheamplitudea (1) andthephaseΔ (1) wheretheparameter k isdeterminedbytheintensityoftheexcitation

9. Forced Duffing Oscillator Theequationandthephaseplaneof LPT

10. Model Formulation System under investigation comprises dimensionless, weakly nonlinear oscillator subject to a two term harmonic excitation in the neighborhood of 1:1 resonance: Y. Starosvetsky, L.I. Manevitch, PHYSICAL REVIEW E 83, 046211 Asymptotical Adoptions:

11. Theoretical Study – Undamped Case Concept of Limiting Phase Trajectories – Analytical Study Performing additional change of variables: ‘Super – Slow Time Scale’ Integral of motion is easily found: Adiabatic Invariant The main goal: Study the relationship between the parameters controlling the global system dynamics

12. Theoretical Study – Undamped Case Concept of Limiting Phase Trajectories – Adiabatic Evolution of LPTs Adiabatic variation of LPTs in super slow time scale Bifurcation of LPT of the first kind and transition to the LPT of the second kind Evolution of LPT of the second kind Evolution of LPT of the first kind Condition for the bifurcation of LPT

13. Theoretical Study – Damped Case Global Dynamics on a Slow Invariant Manifold (SIM) Slow-Flow Model for a Damped Case - Slow Time Scale - Super Slow Time Scale Proceeding with the multi-scale expansion:

14. Theoretical Study – Damped Case Global Dynamics on a Slow Invariant Manifold (SIM) Assuming strong time scales separation , we would like to analyze separately the slow and the super slow evolution of the damped oscillator. Super-Slow Evolution Setting: The evolution on a SIM is governed by: Simple algebraic manipulations will bring us to the following convenient form (Projection of the SIM to the plane )

15. Theoretical Study – Damped Case Global Dynamics on a Slow Invariant Manifold (SIM) Various regimes of a Duffing Oscillator subject to a bi-harmonic forcing Case 1: Only stationary (fixed) points on the stable branches of SIM are possible (Simple Periodic Regimes) Case 2: Case 3: Amplitude of excitation doesn’t reach the fold points which results in permanent, weakly modulated regime Amplitude of excitation exceeds the fold points which results in strongly modulated response, with - relaxations

16. Theoretical Study – Damped Case Global Dynamics on a Slow Invariant Manifold (SIM) Sufficient conditions for Relaxation Oscillations Fold points ‘Case 2’ ‘Case 3’

17. Highly Degenerate System Is the ideology of LPT valid for a strongly degenerate case? Excited oscillator ! Note: System is governed by a single parameter • Main objectives: • Conditions for energy localization • Conditions for recurrent energy exchanges between the oscillators

18. Highly Degenerate System The answer is: Yes! 1. Energy Localization on the first oscillator:

19. Highly Degenerate System 1:1 Resonance 2. Strong Energy Exchanges between the oscillators:

20. Highly Degenerate System Regime of strong localization Assume that energy is permanently localized on the first oscillator: Master and Slave:

21. Highly Degenerate System Solution for a Master Equation Slave Equation: Forced Cubic Oscillator Rescaling: Seek for the strongly localized regime in the form:

22. Highly Degenerate System Multi-Scale Expansion Looking for periodic regimes, we set: Next order approximation: Direct expansion: Fast Components

23. Highly Degenerate System Averaging with respect to a fast time scale, yields Localized solution, yields

24. Highly Degenerate System Linear stability analysis, of the localized solution: Quasi-Periodic Parametric Excitation Stability Analysis - Crude Approximation 1. BroerH.W., Puig, J. and Sim´o, C., Resonance tongues and instability pockets in the quasi-periodic Hill-Schr¨odinger equation, Commun. Math. Phys. 241, pp. 467-503, 2003. 2. Zounes, R.S. and Rand, R.H., Transition curves for the quasi-periodic Mathieu equation, SIAM J. Appl. Math. 58, pp. 1094-1115, 1998

25. Highly Degenerate System Linear stability analysis, of the localized solution: Applying the method of strained parameters: Seeking for intersection with the first tongue: Proceeding with the analysis, yields:

26. Highly Degenerate System Linear stability analysis, of the localized solution: Boundary of the first tongue: For the original equation: Conditions for a critical value of

27. Energy Exchanges and Localization in Highly Degenerate Scalar Models Weakly coupled chains (k<<1): Wire Wire

28. Energy Exchanges and Localization in Highly Degenerate Scalar Models Weakly coupled chains (k<<1): Excitation of spatially periodic and localized modes on a single chain: 1. Anti-Phase Mode

29. Energy Exchanges and Localization in Highly Degenerate Scalar Models Below the threshold:

30. Energy Exchanges and Localization in Highly Degenerate Scalar Models Above the threshold:

31. Energy Exchanges and Localization in Highly Degenerate Scalar Models Weakly coupled chains (k<<1): Excitation of spatially periodic and localized modes on a single chain: 2. Strong localization of compactons Y.S. Kivshar,’Intrinsic Localized Modes as Solitons with Compact Support’, Phys. Rev. E., Vol. 48, 1 (1993)

32. Energy Exchanges and Localization in Highly Degenerate Scalar Models Weakly coupled chains (k<<1): (Above a Certain Threshold) Excitation of spatially periodic and localized modes on a single chain: 2. Strong localization of compactons Y.S. Kivshar,’Intrinsic Localized Modes as Solitons with Compact Support’, Phys. Rev. E., Vol. 48, 1 (1993)

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