Preprint of the project: miscellaneous activities
06/02 Vera Thümmler.
Numerical bifurcation analysis of relative equilibria with Femlab
Relative equilibria are special solutions of partial differential
equations (PDEs), which are stationary in an appropriate comoving frame of reference.
Such solutions occur frequently in biological and chemical models, e.g.
when describing pattern formation of reaction-diffusion equations.
Examples are traveling waves in 1d, planar and spiral waves in 2d and
scroll waves in 3d. If the equation has a special symmetry property - equivariance, then
one can transform the equation into the comoving frame during the computation
of the solution. We will show that this can be very convenient for numerical
computations. We have implemented the resulting `frozen equation'' in Femlab and
conducted several numerical experiments for different examples in 1d and 2d such
as traveling waves and spirals in the FitzHugh-Nagumo and the complex Ginzburg-Landau
system. Moreover, we describe some of the problems which arise because of the
additional convective terms which have been introduced by the freezing transformation.
