Scientific Computing Seminar

Date:
Monday, November 1, 2004
Time:
11:00am-12:00pm
Location:
50D-3416
Seminar Speaker:
Vanessa Lopez
Department of Computer Science
University of Illinois at Urbana-Champaign
Title:
Periodic Solutions of 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 that are periodic in time, up to a transformation by an element of the equations' symmetry group. As a model problem we work with the 1D complex Ginzburg-Landau equation (CGLE), which is a standard example of an evolution equation that exhibits chaotic behavior. The problem of finding relative time-periodic solutions numerically 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 as an unknown the group element that defines a relative time-periodic solution. Using this approach, we found a large collection of distinct relative time-periodic solutions in a chaotic region of the CGLE. These solutions, all of which have broad temporal and spatial spectra, were previously unknown. There is a great deal of variety in their Lyapunov spectra and spatio-temporal profiles. Moreover, none bear resemblance to the time-periodic solutions of the CGLE studied previously. We also consider the Navier-Stokes equations for an incompressible fluid and present preliminary work towards the problem of finding relative time-periodic solutions of these equations.
Sponsor of Seminar:
Ali Pinar
Scientific Computing

Contact Esmond G. Ng EGNg@lbl.gov