| 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.
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