On forced oscillations in groups of interacting nonlinear systems

TL;DR

This paper establishes conditions for the existence of periodic solutions in complex systems composed of interacting nonlinear subsystems, demonstrated through examples like coupled pendulums in a periodic field.

Contribution

It provides new sufficient conditions for periodic solutions in large, interacting nonlinear systems with external periodic forcing.

Findings

Conditions ensure existence of periodic solutions

Application to coupled pendulums example

Framework applicable to various nonlinear systems

Abstract

Consider a periodically forced nonlinear system which can be presented as a collection of smaller subsystems with pairwise interactions between them. Each subsystem is assumed to be a massive point moving with friction on a compact surface, possibly with a boundary, in an external periodic field. We present sufficient conditions for the existence of a periodic solution for the whole system. The result is illustrated by a series of examples including a chain of strongly coupled pendulums in a periodic field.