Non-stationary resonance dynamics of weakly coupled pendula

TL;DR

This paper introduces a unified framework using Limiting Phase Trajectories to analyze non-stationary resonance dynamics in weakly coupled pendula, revealing energy exchange, localization, and chaos across all initial conditions.

Contribution

It presents a novel, comprehensive approach to understanding resonance dynamics without amplitude restrictions, extending analysis to all initial conditions.

Findings

Conditions for energy exchange and localization identified

Roots and domains of chaos clarified

Analytical results confirmed by numerical simulations

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 super- conducting 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 corrobo- rated 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.