Nonstationary dynamics of the sine lattice consisting of three pendula (trimer).

Abstract

The low- and high-amplitude oscillations in the system of three nonlinear coupled pendula (trimer) are analyzed beyond the quasilinear approximation. The considered oscillations are fundamental for many models of the energy exchange processes in physical, mechanical, and biological systems, in particular, for the torsional vibrations of flexible polymers or DNA's double strands. We obtained the conditions of the basic stationary solutions' stability. These solutions correspond to the nonlinear normal modes (NNMs), the instability of which leads to the appearance of localized NNMs (stationary energy localization). Using an asymptotic procedure, we reduce the dimension of the system's phase space that allows us to analyze the energy exchange between pendula in the slow timescale and to reveal periodic interparticle energy exchange and nonstationary energy localization. It has been shown recently that essentially nonstationary resonance processes of this type are adequately described in terms of the limiting phase trajectories (LPTs) corresponding to beatings between the oscillators or coherence domains in the slow timescale. Moreover, it turns out that criteria of the transition to the stationary and nonstationary energy localization can be formulated as the bifurcation conditions for NNMs and LPTs, respectively. The trimer under consideration is a nonintegrable system, and therefore its equations of motion is only after dimensions reduction can be analyzed by the Poincare sections method. Finally, we aim to study the highly nonstationary regimes, which correspond to beatinglike periodic or quasiperiodic recurrent energy exchange between the pendula.