Degree Theory and Symmetric Equations Assisted by GAP System: With a Special Focus on Systems with Hysteresis(Mathematical Surveys and Monographs)

Symmetries are a common feature of real-world phenomena in many fields, including physics, biology, materials science, and engineering. They can help understand the behavior of a system and optimize engineering designs. Nonlinear effects such as delays, nonsmoothness, and hysteresis can have a significant impact on the dynamics and contribute to the increased complexity of symmetric systems. The goal of this book is to provide a complete theoretical and practical manual for studying a large class of dynamical problems with symmetries using degree theory methods. To study the impact of symmetries on the occurrence of periodic solutions in dynamical systems, special variants of the Brouwer degree, the Brouwer equivariant degree, and the twisted equivariant degree are developed to predict patterns, regularities, and symmetries of solutions. Applications to specific dynamical systems and examples are supported by a software package integrated with the GAP system, which provides assistance in the group-theoretic computations involved in equivariant analysis. This book is intended for readers with a basic knowledge of analysis and algebra, including researchers in pure and applied mathematical analysis, graduate students, and scientists interested in areas involving mathematical modeling of symmetric phenomena. The text is self-contained, and the necessary background material is provided in the appendices.

Table of Contents:
Introduction Brouwer equivariant degree and applications Local Brouwer degree Equivariant Brouwer degree Subharmonic solutions to reversible difference equations Periodic solutions to $\kappa$-reversible continuous time systems with multiple delays Equivariant bifurcation of periodic solutions with fixed period Non-radial solutions to coupled semilinear elliptic systems on a disc Twisted equivariant degree and applications Local $S^1$-equivariant degree Local twisted equivariant degree Two parameter $G$-equivariant bifurcation Hopf bifurcation Hopf bifurcation of relative periodic solutions Global Hopf bifurcation of differential equations with threshold type state-dependent delay by Qingwen Hu Hysteresis models as rate-independent operators Hopf bifurcations in systems of symmetrically coupled oscillators with hysteretic elements Hopf bifurcation in nonlinear parabolic equations Appendices Elements of differential topology Lie groups and their topological actions Elements of representation theory $G$-manifolds and smooth $G$-vector bundles Amalgamated notation Quickstart for GAP and EquiDeg Bibliography Index