Stability of a Parametrically Driven, Coupled Oscillator System: An Auxillary Function Method Approach

TL;DR

This paper introduces a novel auxiliary function method combined with classical analysis to accurately determine the stability regions of a parametrically driven, coupled oscillator system, especially away from resonance frequencies.

Contribution

The paper presents a new auxiliary function method for long-time averages, providing a robust and efficient way to compute stability boundaries in coupled oscillators, surpassing traditional asymptotic methods.

Findings

High order coupling effects can cause asymptotic methods to be dangerously un-conservative.

The auxiliary function method accurately predicts true stability regions across all initial conditions.

Differences between true and approximate stability boundaries are significant at relevant parameter values.

Abstract

Coupled, nonlinear oscillators are often studied in applied biology, physics, fluids, and many other disciplines. In this paper, we study a parametrically driven, coupled oscillator system where the individual oscillators are subjected to varying frequency and phase with a focus on the influence of the damping and coupling parameters away from parametric resonance frequencies. In particular, we study the key long-term statistics of the oscillator system's trajectories and stability. We present a novel, robust and computationally efficient method come to be known as an auxillary function method for long-time averages, and we pair this method with classical, perturbative-asymptotic analysis to corroborate the results of this auxillary function method. These paired methods are then used to compute the regions of stability for a coupled oscillator system. The objective is to explore the influence of higher order, coupling effects on the stability boundary across a broad range of modulation frequencies, including frequencies away from parametric resonances. We show that both simplified and more general asymptotic methods can be dangerously un-conservative in predicting the true regions of stability due to high order effects caused by coupling parameters. The differences between the true stability boundary and the approximate stability boundary can occur at physically relevant parameter values in regions away from parametric resonance. The differences between the solutions depends on the specific parameters of the system, as explained in the results section. As an alternative to asymptotic methods, we show that the auxillary function method for long-time averages is an efficient and robust means of computing true regions of stability across all possible initial conditions.