Singularity Theory and Equivariant Symplectic Maps

The monograph is a study of the local bifurcations of multiparameter symplectic maps of arbitrary dimension in the neighborhood of a fixed point.The problem is reduced to a study of critical points of an equivariant gradient bifurcation problem, using the correspondence between orbits ofa symplectic map and critical points of an action functional. New results onsingularity theory for equivariant gradient bifurcation problems are obtained and then used to classify singularities of bifurcating period-q points. Of particular interest is that a general framework for analyzing group-theoretic aspects and singularities of symplectic maps (particularly period-q points) is presented. Topics include: bifurcations when the symplectic map has spatial symmetry and a theory for the collision of multipliers near rational points with and without spatial symmetry. The monograph also includes 11 self-contained appendices each with a basic result on symplectic maps. The monograph will appeal to researchers and graduate students in the areas of symplectic maps, Hamiltonian systems, singularity theory and equivariant bifurcation theory.

Generic bifurcation of periodic points.- Singularity theory for equivariant gradient bifurcation problems.- Classification of Zq-equivariant gradient bifurcation problems.- Period-3 points of the generalized standard map.- Classification of Dq-equivariant gradient bifurcation problems.- Reversibility and degenerate bifurcation of period-q points of multiparameter maps.- Periodic points of equivariant symplectic maps.- Collision of multipliers at rational points for symplectic maps.- Equivariant maps and the collision of multipliers.