Symmetry and chaos in the complex Ginzburg–Landau equation. II. Translational symmetries

The complex Ginzburg–Landau (CGL) equation on a one-dimensional domain with periodic boundary conditions has a number of different symmetries, and solutions of the equation may or may not be fixed by the action of these symmetries. We investigate the stability of chaotic solutions that are spatially periodic with period L with respect to subharmonic perturbations that have a spatial period kL for some integer k>1. This is done by considering the isotypic decomposition of the space of solutions and finding the dominant Lyapunov exponent associated with each isotypic component. We find a region of parameter space in which there exist chaotic solutions with spatial period L and homogeneous Neumann boundary conditions that are stable with respect to perturbations of period 2L. On varying the parameters it is possible to arrange for this solution to become unstable to perturbations of period 2L while remaining chaotic, leading to a supercritical subharmonic blowout bifurcation. For a large number of parameter values checked, chaotic solutions with spatial period L were found to be unstable with respect to perturbations of period 3L. We conclude that while periodic boundary conditions are often convenient mathematically, we would not expect to see chaotic, spatially periodic solutions forming starting with an arbitrary, non-periodic initial condition.