A recent development of the equivariant degree methods and their applications in symmetric dynamical systems.

摘要

This work consists of three parts: theoretical preparations, applications and computational results. The theoretical part primarily addresses the basic properties and computational scheme of various equivariant degrees, including the general equivariant degree, primary equivariant degree, twisted primary degree, S1-degree and equivariant gradient degree (Part I). The second part contains two types of applications of equivariant degree methods in the area of equivariant nonlinear analysis: the symmetric Hopf bifurcation and the existence of periodic solutions in autonomous symmetric systems (Part II). The last part presents an appendix of Sobolev spaces and a catalogue for several groups (their subgroups, irreducible representations and basic degrees), multiplication tables and computational results obtained throughout the thesis (Part III).