Variety of rotation modes in a small chain of coupled pendulums

Abstract

This article studies the rotational dynamics of three identical coupled pendulums. There exist two parameter areas where the in-phase rotational motion is unstable and out-of-phase rotations are realized. Asymptotic theory is developed that allows to analytically identify boarders of instability areas of in-phase rotation motion. It is shown that out-of-phase rotations are the result of parametric instability of in-phase motion. Complex out-of-phase rotations are numerically found and their stability and bifurcations are defined. It is demonstrated that emergence of chaotic dynamics happens due period doubling bifurcation cascade. The detail scenario of symmetry breaking is presented. The development of chaotic dynamics leads to origin of two chaotic attractors of different types. The first one is characterized by the different phases of all pendulums. In the second case the phases of two pendulums are equal, and the phase of the third one is different. This regime with partial symmetry breaking is a chaotic chimera.