Thomas J. Bridges, Jacques E. Furter Singularity Theory and Equivariant Symplectic Maps

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The monograph is a study of the local bifurcations ofmultiparameter symplectic maps of arbitrary dimension in theneighborhood of a fixed point.The problem is reduced to astudy of critical points of an equivariant gradientbifurcation problem, using the correspondence between orbitsofa symplectic map and critical points of an actionfunctional. New results onsingularity theory forequivariant gradient bifurcation problems are obtained andthen used to classify singularities of bifurcating period-qpoints. Of particular interest is that a general frameworkfor analyzing group-theoretic aspects and singularities ofsymplectic maps (particularly period-q points) is presented. Topics include: bifurcations when the symplectic map hasspatial symmetry and a theory for the collision ofmultipliers near rational points with and without spatialsymmetry. The monograph also includes 11 self-containedappendices each with a basic result on symplectic maps. Themonograph will appeal to researchers and graduate studentsin the areas of symplectic maps, Hamiltonian systems,singularity theory and equivariant bifurcation theory.