Stationary and Non-stationary Resonance Dynamics of the Finite Chain of Weakly Coupled Pendula
We discuss new phenomena of energy localization and transition to chaos in the finite system of coupled pendula (which is a particular case of the Frenkel-Kontorova model), without any restrictions on the amplitudes of oscillations. The direct significant applications of this fundamental model comprise numerous physical systems. In the infinite and continuum limit the considered model is reduced to integrable sine-Gordon equation or certain non-integrable generalizations of it. In this limit, the chaotization is absent, and the energy localization is indicated by the existence of soliton-like solutions (kinks and breathers). As for more realistic finite models, analytical approaches are lacking, with the exception of cases limited to two and three pendula. We propose a new approach to the problem based on the recently developed Limiting Phase Trajectory (LPT) concept in combination with a semi-inverse method. The analytical predictions of the conditions providing transition to energy localization are confirmed by numerical simulation. It is shown that strongly nonlinear effects in finite chains tend to disappear in the infinite limit.
CYBERNETICS AND PHYSICS, Vol. 5, No. 4, 2016 , 130–135.