Periodic Solutions of Chaotic Partial Differential Equations with Symmetries

Lopez, Vanessa

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https://hdl.handle.net/2142/10773 Copy

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Title

Periodic Solutions of Chaotic Partial Differential Equations with Symmetries

Author(s)

Lopez, Vanessa

Issue Date

2004-01

Keyword(s)

chaotic partial differential equations

Abstract

"We consider the problem of finding ""relative"" time-periodic solutions of chaotic partial differential equations with symmetries. Relative time-periodic solutions are solutions which are periodic in time, up to a transformation by an element of the equations' symmetry group. First, we work with the complex Ginzburg-Landau equation (CGLE) as a model problem. The problem of finding relative time-periodic solutions is reduced to one of finding solutions to a system of nonlinear algebraic equations, obtained after applying a spectral-Galerkin discretization in space and time to the CGLE. The discretization is designed to include the group element (that defines a relative time-periodic solution) as an unknown. We found a large collection of distinct relative time-periodic solutions in a chaotic region of the CGLE. These solutions were previously unknown and all have broad temporal and spatial spectra. There is a great deal of variety in their Lyapunov spectra and spatio-temporal profiles. In particular, none bear resemblance to the coherent structures or generalized traveling waves widely studied in the literature. The problem of finding relative time-periodic solutions to the 3D Navier-Stokes equations is also considered. As a preliminary step for finding such solutions, we implemented a time integration procedure using a spectral-Galerkin method and trigonometric and Legendre polynomials to generate basis functions. Details about the procedure are discussed. For this problem, the system of nonlinear algebraic equations whose solutions yield the relative time-periodic orbits is very large and its Jacobian matrix is dense. We present the insights gained from our attempts to solve linear systems with the Jacobian as coefficient matrix using iterative methods."

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text

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http://hdl.handle.net/2142/10773

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