Harmonic forcing of an extended oscillatory system: Homogeneous and periodic solutions

TL;DR

This paper investigates how external harmonic forcing influences a one-dimensional oscillatory system modeled by the complex Ginzburg-Landau equation, revealing transitions from homogeneous to stripe states and deriving an analytic phase equation to understand stability.

Contribution

It introduces an approximate phase equation for the forced CGLE and provides an analytic solution for the stripe state, enhancing understanding of stability borders.

Findings

Homogeneous state becomes unstable to stripe state as forcing decreases.

Analytic phase equation explains stability borders.

Stripe state stability is asymmetric.

Abstract

In this paper we study the effect of external harmonic forcing on a one-dimensional oscillatory system described by the complex Ginzburg-Landau equation (CGLE). For a sufficiently large forcing amplitude, a homogeneous state with no spatial structure is observed. The state becomes unstable to a spatially periodic ``stripe'' state via a supercritical bifurcation as the forcing amplitude decreases. An approximate phase equation is derived, and an analytic solution for the stripe state is obtained, through which the asymmetric behavior of the stability border of the state is explained. The phase equation, in particular the analytic solution, is found to be very useful in understanding the stability borders of the homogeneous and stripe states of the forced CGLE.