Symmetry breaking and branching patterns in equivariant bifurcation theory. II (Q1196250)

The authors present applications and examples of the theory developed in part I [ibid. 118, No. 4, 297-348 (1992; Zbl 0760.58008)]. They are mainly interested in subgroups \(G\) of the full symmetry group \(W(B^ k)\) of the \(k\)-dimensional cube or of the symmetric group \(S_{k+1}\). Both, \(W(B^ k)\) and \(S_{k+1}\) act on \(V = \mathbb{R}^ k\) absolutely irreducible. The local bifurcation pattern for solutions of the equation \(f(x)+\lambda x = 0\), where \(f: V \to V\) is \(W(B^ k)\)-equivariant and polynomial, is determined (for generic \(f\)) by the terms of order at most 3. It is shown that this is still true if \(f\) is only \(G\)-equivariant provided the restricted action of \(G\subset W(B^ k)\) on \(V\) is also absolutely irreducible and there are no non-trivial homogeneous quadratic \(G\)-equivariants. In this case it is possible to analyze the possible branching patterns for a generic \(f\). In particular, the authors obtain many new examples where the ``Maximal Isotropy Subgroup Conjecture'' does not hold. One section of the paper is concerned with the dynamics of \(G\)- equivariant bifurcation problems on \(\mathbb{R}^ k\) for certain subgroups \(G\) of \(W(B^ k)\). The authors present examples where branches of heteroclinic cycles and limit cycles exist.