Transition to spatiotemporal chaos via stationary branching shocks and holes

Abstract

Spatiotemporal chaos in the complex Ginzburg–Landau equation is known to be associated with a rapid increase in the density of defects, which are isolated points at which the solution amplitude is zero and the phase is undefined. Recently there have been significant advances in understanding the details and interactions of defects and other coherent structures, and in the theory of convective and absolute stability. In this paper, the authors exploit both of these advances to update and clarify the onset of spatiotemporal chaos in the particular case of the complex Ginzburg–Landau equation with zero linear dispersion. They show that very slow increases in the coefficient of nonlinear dispersion cause a shock–hole (defect) pair to develop in the midst of a uniform expanse of plane wave. This is followed by a cascade of splittings of holes into shock–hole–shock triplets, culminating in spatiotemporal chaos at a parameter value that matches the change in absolute stability of the plane wave. The authors demonstrate a close correspondence between the splitting events and theoretical predictions, based on the theory of absolute stability. They also use measures based on power spectra and spatial correlations to show that when the plane wave is convectively unstable, chaos is restricted to localised regions, whereas it is extensive when the plane wave is absolutely unstable.