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.
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
Zalman Balanov, University of Texas at Dallas, Richardson, TX, Wieslaw Krawcewicz, University of Texas at Dallas, Richardson, TX.
Dmitrii Rachinskii, University of Texas at Dallas, Richardson, TX, Hao-Pin Wu, National Yang Ming Chiao Tung University, Hsinshu, Taiwan.
Jianshe Yu, Guangzhou University, China.
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