Symmetry and chaos in the complex Ginzburg± Landau equationÐ I. Re ̄ ectional symmetries

The complex Ginzburg± Landau (CGL) equation on a one-dimensional domain with periodic boundary conditions has a number of diþ erent symmetries. Solutions of the CGL equation may or may not be ® xed by the action of these symmetries. We investigate the stability of chaotic solutions with some re ̄ ectional symmetry to perturbations which break that symmetry. This can be achieved by considering the isotypic decomposition of the space and ® nding the dominant Lyapunov exponent associated with each isotypic component. Our numerical results indicate that for most parameter values, chaotic solutions that have been restr icted to lie in invariant subspaces are unstable to perturbations out of these subspaces, leading us to conclude that for these parameter values arbitrary initial conditions will generically evolve to a solution with the minimum amount of symmetry allowable. We have also found a small region of parameter space in which chaotic solutions that are even are stable with respect to odd perturbations.