Forced Nonlinear Resonance in a System of Coupled Oscillators

TL;DR

This paper investigates the long-term behavior of a coupled nonlinear oscillator system under periodic forcing, revealing resonance phenomena and deriving an envelope Hamiltonian that matches numerical simulations.

Contribution

It introduces an analytical approach to describe forced nonlinear resonance in coupled oscillators, deriving a Mathieu equation and envelope Hamiltonian for large time scales.

Findings

One component follows an inhomogeneous Mathieu equation.

The other component exhibits large amplitude pulsations.

Analytic results align with numerical simulations.

Abstract

We consider a resonantly perturbed system of coupled nonlinear oscillators with small dissipation and outer periodic perturbation. We show that for the large time $t \sim \epsilon^{-2}$ one component of the system is described in the main by the inhomogeneous Mathieu equation while the other component represents pulsation of large amplitude. A Hamiltonian system is obtained which describes the behaviour of the envelope in the main. The analytic results agree to numerical simulations.