Relative Periodic Solutions of the Cgle

A method of finding relative periodic orbits for differential equations with continuous symmetries is described and its utility demonstrated by computing relative periodic solutions for the one-dimensional complex Ginzburg-Landau equation (CGLE) with periodic boundary conditions. A relative periodic solution is a solution that is periodic in time, up to a transformation by an element of the equation's symmetry group. With the method used, relative periodic solutions are represented by a space-time Fourier series modified to include the symmetry group element and are sought as solutions to a system of nonlinear algebraic equations for the Fourier coefficients, group element, and time period. The 77 relative periodic solutions found for the CGLE exhibit a wide variety of temporal dynamics, with the sum of their positive Lyapunov exponents varying from 5.19 to 60.35 and their unstable dimensions from 3 to 8. Preliminary work indicates that weighted averages over the collection of relative periodic solutions accurately approximate the value of several functionals on typical trajectories. 1. Introduction. The complex Ginzburg-Landau equation (CGLE) is a widely studied partial differential equation with applications in many areas of science. (See [1, 44] and references therein.) It has become a model problem for the study of nonlinear evolution equations with chaotic spatio-temporal dynamics. In this paper we work with the CGLE with cubic nonlinearity in one spatial dimension,