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Limiting Phase Trajectories: a New Paradigm for the Study of Highly Non-Stationary Processes in Nonlinear Physics

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Limiting Phase Trajectories: a New Paradigm for the Study of Highly Non-Stationary Processes in Nonlinear Physics Limiting Phase Trajectories: a new paradigm for the study of highly non-stationary processes in Nonlinear Physics Leonid I. Manevitcha, Agnessa Kovalevab1, Yuli Starosvetskyc aInstitute of Chemical Physics, Russian Academy of Sciences, Moscow 117997, Russia bSpace Research Institute, Russian Academy of Sciences, Moscow 119991, Russia cTechnion -Israel Institute of Technology,Technion City, Haifa 32000, Israel This Report discusses a recently developed concept of Limiting Phase Trajectories (LPTs) providing a unified description of resonant energy transport in a wide range of classical and quantum dynamical systems with constant and time-varying parameters. It is shown that strongly modulated non- stationary processes occurring in a nonlinear oscillator array under certain initial conditions may characterize either maximum possible energy exchange between the clusters of oscillators (effective particles) or maximum energy transfer from an external source of energy to the system. The trajectories corresponding to these processes are referred to as LPTs. The development and the use of the LPT concept are motivated by the fact that highly non-stationary resonance processes occurring in a broad variety of finite-dimensional physical models are beyond the well-known paradigm of the nonlinear normal modes (NNMs), fully justified only for stationary processes, their stability and bifurcations, as well as for non-stationary processes described approximately by some combinations of non-resonating normal modes. Thus, the role of LPTs in understanding and description of non- stationary energy transfer is similar to the role of NNMs for the stationary processes. Several applications of the LPT concept to significant nonlinear problems and a scenario of the transition from regular to chaotic behavior with the LPTs implication are presented. In order to highlight the novelty and perspectives of the developed approach, we place the LPT concept into the context of complex dynamical phenomena related to energy transfer problems. Key words: limiting phase trajectories, nonlinear normal modes, non-stationary resonance processes, energy exchange, localization, regular motion, chaotic motion. *Corresponding author. Tel.:+7 916 9824740; fax: +7 495 913 3040. E-mail address:[email protected] (A. Kovaleva). 1 Content 1. Introduction 1.1. Nonlinear normal modes in finite oscillatory chains 1.2. Solitonic excitations in the infinite models 1.3. Discrete breathers 1.4. Topological solitons 1.5. LPTs of resonant non-stationary processes in a finite chain: a preliminary consideration 1.5.1. LPTs in a symmetric system of two identical oscillators 1.5.2. Non-smooth transformations 1.5.3. Weakly dissipative systems 1.6. Further extensions of the LPT theory 1.6.1. Linear classical and quantum systems 1.6.2. Nonlinear infinite-dimensional models 1.6.3. Autoresonance 1.6.4. Targeted energy transfer 1.6.5. Synchronization of autogenerators 2. Limiting Phase Trajectories of a single oscillator 2.1. Duffing oscillator with harmonic forcing near 1:1 resonance 2.1.1. Formulation of the problem and main equations 2.1.2. Stationary states, LPTs, and critical parameters of non-dissipative systems 2.1.3. Non-smooth approximations 2.1.4. Transient dynamics of dissipative systems 2.2. Duffing oscillator subject to biharmonic forcing near the primary resonance 2.2.1. Equations of fast and slow motion 2.2.2. LPTs of slow motion in a non-dissipative system 2.2.3. Super-slow model 2.2.4. Relaxation oscillations in a lightly-damped system 3. Limiting Phase Trajectories of weakly coupled nonlinear oscillators. Classical-quantum analogy 3.1. Resonance energy transfer in a 2DOF system 3. 2. Stationary points and LPTs 3.3. Critical parameters 4. Weakly coupled oscillators under the condition of sonic vacuum 4.1. Evidence of energy localization and exchange in coupled oscillators in the state of sonic vacuum 4.2. Asymptotic analysis of resonance motion 2 4.2.1. Fixed points and NNMs in the neighborhood of resonance 4.2.2. Limiting Phase Trajectories 4.3. Numerical analysis of the fundamental model 5. Non-conventional synchronization of weakly coupled active oscillators (autogenerators) 5.1. Lienard oscillators 5.2. Coupled active oscillators 5.3. NNMs and LPTs 5.4. Analysis of the phase plane and analytical solutions 6. Limiting phase trajectories in passage through resonance in time-variant systems 6.1. Autoresonance in a SDOF nonlinear oscillator 6.1.1. Critical parameters 6.1.2. Numerical study of capture into resonance 6.2. Autoresonance versus localization in weakly coupled oscillators 6.2.1. Energy transfer in a system with constant excitation frequency 6.2.2. Energy localization and transfer in a system with a slowly-varying forcing frequency 6.2.3. Energy transfer in a system with slowly-varying natural and excitation frequencies 6.3. Classical analog of linear and quasi-linear quantum tunneling 6.3.1. Two weakly coupled linear oscillators 6.3.2. Approximate analysis of energy transfer in the linear system 6.3.3 Classical and quantum tunneling in coupled linear oscillators 6.4. Moderately and strongly nonlinear adiabatic tunneling 6.4.1. Moderately nonlinear regimes 6.4.2. Strongly nonlinear regimes 7. Energy exchange, localization and transfer in finite oscillatory chains. Brief review 7.1. Bifurcations of LPTs and routes to chaos in an anharmonic 3DOF chain 7.1.1. Equations of the slow flow model in a 3DOF chain 7.1.2. Nonlinear Normal Modes 7.1.3. Emergence and bifurcations of LPTs in the chain of three coupled oscillators 7.1.4. Numerical results 7.1.5. Spatially localized pulsating regimes 7.2. Higher-dimensional oscillator chains: Brief Discussion 8. Conclusions Acknowledgements References 3 1. Introduction This Report presents the concept of Limiting Phase Trajectories (LPTs), depicts their intrinsic properties, and demonstrates the role of LPTs in intense resonance energy transfer in physical systems of different nature. By definition, motion along the LPT corresponds either to maximum possible energy exchange between the clusters of oscillators (effective particles) or to maximum energy transfer from an external source of energy to the system. In both cases, the system under consideration exhibits strongly modulated non-stationary processes. The derivation of explicit analytical solutions describing non-stationary dynamics of nonlinear systems is, in general, a daunting task. The well-developed techniques of Nonlinear Normal Modes (NNMs), which is effective in the analysis of stationary or non-stationary non-resonant oscillations, cannot be applied to the study of highly non-stationary resonant energy exchange between nonlinear oscillators or clusters of oscillators (see Sections 1.1, 7.2). Recent authors’ works have resolved this difficulty by introducing the notion of Limiting Phase Trajectory, which is an alternative to Nonlinear Normal Mode. Convenient asymptotic procedures for constructing explicit approximations of the LPTs have been suggested. In this introductory part we briefly describe main approaches to the study of nonlinear interactions between oscillators and present an example, which gives a preliminary idea about the LPTs, their properties and advantages in the study of resonance energy transport. A more detailed study of LPTs in systems with one and two degrees of freedom is postponed until Sections 2 and 3. Sections 4 - 7 demonstrate extensions and applications of the LPT concept to the analysis of resonance energy transport in physical systems of different nature, and the role of LPTs in the transition from regular to chaotic behavior. A comprehensive exposition of the reported results can be found in the works cited in this review. 4 1.1. Nonlinear normal modes in finite oscillatory chains Non-stationary resonance processes are of the great interest in various fields of modern physics and engineering dealing with energy transfer and exchange. However, their particularities have not been fully recognized for a long time because of the domination of the common idea about the possible representation of any non-stationary process as a combination of stationary oscillations or waves. Actually, in the linear theory the superposition principle implies the representation of every dynamical process as a sum of normal vibrations or normal waves. This representation is approximately valid even in the nonlinear case in the absence of the intermodal resonance. Thus, main conventional analytical tools for the study of stationary and non-stationary nonlinear processes, their stability and bifurcations in finite dimensional systems have been associated with the NNM concept. Rosenberg [157-159] defined a NNM of an undamped discrete multi-particle system as a synchronous periodic oscillation wherein the displacements of all material points of the system reach their extreme values and pass through zero simultaneously. Then the NNM concept was extended to finite continuum models [202], to systems subjected to external forcing (e.g., [116, 202] and references therein), as well as to auto-generators, for which NNM can be considered as an attractor corresponding to synchronized motion of oscillators [152]. An important feature that distinguishes NNMs from their linear counterpart, besides an amplitude dependence of the frequency, is that they can exceed in number the degrees of freedom of an oscillatory system. In this case, essentially nonlinear localized NNMs are generated through NNM bifurcations, breaking the symmetry of the dynamics and resulting
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