Non-stationary resonance dynamics of weakly coupled pendula

Abstract

In this paper we fill the gap in understanding the non-stationary resonance dynamics of the weakly coupled pendula model, having significant applications in numerous fields of physics such as superconducting Josephson junctions, Bose-Einstein condensates, DNA, etc.. While common knowledge of the problem is based on two alternative limiting asymptotics, namely the quasi-linear approach and the approximation of independent pendula, we present a unified description in the framework of new concept of Limiting Phase Trajectories (LPT), without any restriction on the amplitudes of oscillation. As a result the conditions of intense energy exchange between the pendula and transition to energy localization are revealed in all possible diapason of initial conditions. By doing so, the roots and the domain of chaotic behavior are clarified as they are associated with this transition while simultaneously approaching the pendulum separatrix. The analytical findings are corroborated by numerical simulations. By considering the simplest case of two weakly coupled pendula, we pave the ground for new opening possibilities of significant extensions in both fundamental and applied directions. [7]. The majority of the results in all these fields relate to stationary dynamics and are based on the fundamental stationary regimes, namely the Nonlinear Normal Modes (NNMs) in finite systems [8,9] and solitons (breathers) in infinite models [1,2]. As for non-stationary processes, NNMs can also be used for their description provided that the intermodal resonance is absent [10]. However, the non-stationary resonance dynamics of finite systems turns out to be much more complicated. Therefore only isolated and predominantly numerical results were obtained in this field [11,12]. The recently developed concept of Limiting Phase Trajectories (LPTs) allowed for a systematic approach to description of non-stationary resonance regimes [13][14][15][16][17][18], including coupled pendula dynamics [19]. This concept introduces a fundamental non-stationary process of new type which corresponds to maximum possible energy exchange between the oscillators (in particular, pendula) or clusters of oscillators. In essence, the LPTs play in the non-stationary resonance dynamics of finite systems the role similar to that of the NNMs in the stationary theory and in the study of non-stationary yet non-resonant regimes. In terms of LPT, the transition from intense energy exchange between some clusters of oscillators (coherence domains), in particular, weakly coupled pendula, to energy localization in the initially excited cluster (oscillator) can also be predicted [11,20,21].However, all existing analytical results in the nonstationary resonance dynamics of finite-dimensional systems relate to some asymptotic limits which are either quasi-linear systems or separated oscillators (pendula) [11,14]. In this study we remove these restrictions and admit arbitrary oscillation amplitude in the framework of LPT concept. Assuming weak coupling, only the closeness to inter-...

Complex Envelope Variable Approximation in Nonlinear Dynamics

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Rigid body stabilization under time-varying perturbations with zero mean values

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