Relative Periodic Solutions of the Complex Ginzburg-Landau Equation

Abstract

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.

32 pages, 12 figures

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• Variational method for locating invariant tori
• Exact Coherent Structures and Chaotic Dynamics in a Model of Cardiac Tissue
• Numerical Continuation of Invariant Solutions of the Complex Ginzburg-Landau Equation
• On the use of stabilising transformations for detecting unstable periodic orbits in the Kuramoto-Sivashinsky equation
• Canonical methods of constructing invariant tori by phase-space sampling
• Minimal cover of high-dimensional chaotic attractors by embedded recurrent patterns
• Efficient method for detection of periodic orbits in chaotic maps and flows