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.
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.
Series: Lecture Notes in Mathematics
Number Of Pages: 230
Published: 29th November 1993
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.39 x 15.6
Weight (kg): 0.34