Global bifurcation of periodic solutions with symmetry / Bernold Fiedler

The author investigates the existence of time-periodic solutions of nonlinear autonomous dynamical systems. Specifically, he considers dynamical systems with symmetry and looks for periodic solutions with prescribed symmetry, such as concentric, or rotating, or discrete waves in applications. One key tool is the design of an index which evaluates changes of stability in such a way that if the index is nonzero, then global Hopf bifurcation occurs with possible symmetries. Generic equivariant approximations to the original problem are also used. Some applications to coupled oscillators and reaction-diffusion systems are discussed, as well as flexible tests which detect oscillations. The main generic and general global results are concentrated in § 2, where the adequate global equivariant Hopf index is defined. The basic concepts of proof are explained in § 3, in case of no symmetry. The idea of virtual symmetry, also crucial in this book, is discussed next (§ 4). Then generic local theory (§ 5), generic global theory (§ 6), general global theory (§ 7), and applications (§ 8) are set up. (Zentralblatt)