Nonlinear Sciences – Chaotic Dynamics
Scientific paper
2004-08-09
SIAM J. Applied Dynamical Systems, Vol. 4, No. 4, November 2005, pages 1042-1075
Nonlinear Sciences
Chaotic Dynamics
32 pages, 12 figures
Scientific paper
10.1137/040618977
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.
Boyland Philip
Heath Michael T.
López Vanessa
Moser Robert D.
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