Variety of rotation modes in a small chain of coupled pendulums

TL;DR

This paper investigates the complex rotational behaviors and stability of a small chain of coupled pendulums, revealing conditions for various synchronized and chaotic states, including the emergence of chaotic chimeras.

Contribution

It develops an asymptotic theory to analytically identify instability borders and explores the bifurcation scenarios leading to chaos and symmetry breaking in coupled pendulums.

Findings

Identification of parameter regions with unstable in-phase rotation

Discovery of out-of-phase rotational modes and their stability

Observation of chaotic dynamics and chimera states

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.