The book has two aims. One is to expound the mathematical methods of equivariant bifurcation theory. Beyond the classical bifurcation tools, such as center manifold and normal form reductions, the presence of symmetry requires the introduction of the algebraic and geometric formalism of Lie group theory and transformation group methods. For the first time, all these methods in equivariant bifurcations are presented in a coherent and self-consistent way in a book.

The other aim is to present the most recent ideas and results in this theory, in relation to applications. This includes bifurcations of relative equilibria and relative periodic orbits for compact and noncompact group actions, heteroclinic cycles and forced symmetry-breaking perturbations. Although not all recent contributions could be included and a choice had to be made, a rather complete description of these new developments is provided. At the end of every chapter, exercises are offered to the reader.

Contents:

• Symmetries in ODE's and PDE's
• Equivariant Bifurcations, A First Look
• Invariant Manifolds and Normal Forms
• Linear Lie Group Actions
• The Equivariant Structure of Bifurcation Equations
• Reduction Techniques for Equivariant Systems
• Relative Equilibria and Relative Periodic Orbits
• Bifurcations in Equivariant Systems
• Heteroclinic Cycles
• Perturbation of Equivariant Systems

"This book is a nice exposition of the basic material related to bifurcations and dynamical systems with symmetry ... The theory is presented in a very clear form to non-specialists, with many examples and a collection of exercises found at the end of each chapter."