Reiner Lauterbach

Methods in Equivariant Bifurcations and Dynamical Systems

Symmetries in ODEs and PDEs equivariant bifurcations, a first look invariant manifolds and normal forms linear Lie group actions the equivariant structure of bifurcation equations reduction techniques for equivariant systems relative equilibria and relative periodic orbits bifurcations in equivariant systems heteroclinic cycles perturbation of equivariant systems.

Symmetry-breaking at non-positive solutions of semilinear elliptic equations

We consider symmetry-breaking bifurcations at non-positive, radially symmetric solutions of semilinear elliptic equations on a ball with Dirichlet boundary conditions. For nonlinearities which are asymptotically affine linear, we find solutions at which the symmetry breaks. The kernel of the linearized equation at these solutions is an absolutely irreducible representation of the group O(n). For this kind of equation a transversality condition is satisfied if the perturbation of the affine linear problem is small enough. Thus we obtain, by the equivariant branching lemma, a large variety of isotropy subgroups of O(n) which occur as symmetries of the bifurcating solution branches.

Exclusion of Relative Equilibria

We study the question of existence of relative equilibria in a SO(3)- equivariant dynamical system when the dynamics is governd by the l=4 representation of this group. This answers a question which remained open in Chossat et al. [1].