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Relative Periodic Solutions of the Complex Ginzburg-Landau Equation | | Vanessa López
; Philip Boyland
; Michael T. Heath
; Robert D. Moser
; | | Date: |
9 Aug 2004 | | Subject: | Chaotic Dynamics | nlin.CD | | 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. | | Source: | arXiv, nlin.CD/0408018 | | Services: | Forum | Review | PDF | Favorites | |
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