Stability of a 2-dimensional Mathieu-type system with quasiperiodic coefficients

TL;DR

This paper investigates the stability of a 2D quasiperiodic Mathieu-type system, revealing complex resonance phenomena, stability islands, and instability arcs through numerical, Floquet, harmonic balance, and multiple scales analyses.

Contribution

It extends Mathieu equations to higher dimensions with quasiperiodic terms and analyzes stability and resonance effects using multiple analytical and numerical methods.

Findings

Identification of stability and instability regions in parameter space.

Discovery of stability islands within instability zones.

Prediction of resonance curves using harmonic balance.

Abstract

In the following we consider a 2-dimensional system of ODE's containing quasiperiodic terms. The system is proposed as an extension of Mathieu-type equations to higher dimensions, with emphasis on how resonance between the internal frequencies leads to a loss of stability. The 2-d system has two `natural' frequencies when the time dependent terms are switched off, and it is internally driven by quasiperiodic terms in the same frequencies. Stability charts in the parameter space are generated first using numerical simulations and Floquet theory. While some instability regions are easy to anticipate, there are some surprises: within instability zones small islands of stability develop, and unusual `arcs' of instability arise also. The transition curves are analyzed using the method of harmonic balance, and we find we can use this method to easily predict the `resonance curves' from which bands of instability emanate. In addition, the method of multiple scales is used to examine the islands of stability near the 1:1 resonance.