Bifurcation and symmetry-breaking (Q2640956)

This paper contains an extended study of the set of solutions of an equation \(M(u,\lambda)=0\), when M is equivariant with respect to a compact Lie group G, in the sense that \(M(gu,\lambda)=gM(u,\lambda)\) (\(\forall)\) \(g\in G\). The authors investigate under what conditions there exists bifurcation from a curve of invariant (under G) solutions to non- invariant solutions. Considering that the u-derivative of M has a finite positive spectrum, they construct a Lyapunov-Schmidt-type reduction and explicitly compute the Conley indices. Applications of the abstract results are presented. The existence of solutions having particular symmetry groups is discussed.