The symmetry perspective. From equilibrium to chaos in phase space and physical space
This skillfully written monograph has been awarded the Ferran Sunyer i Balaguer 2001 prize for a mathematical text of expository nature. Its principal goal is to demonstrate the important role of symmetry in pattern-formation and the ways symmetry can be used for exploring mechanisms by which patterns form. The book is divided into ten chapters and concludes with the exhaustive list of 528 references. The first chapter introduces two important notions: symmetry of a system of ODEs and symmetry of a solution of a system of ODEs which, in general, need not be the same as the symmetry of the system. Then the phenomenon of symmetry-breaking in which solutions have less symmetry than equations is considered. Two key results, Lyapunov-Schmidt reduction which reduces the dimension of a bifurcation problem, and the Equivariant Branching Lemma which provides sufficient conditions for the existence of a bifurcating branch of equilibria with a particular type of symmetry group, called axial, are presented. Then this lemma is applied to the specification model, and a number of important issues on the relevance of symmetry assumptions in modeling are discussed. Chapter 2 deals with the stability of equilibria, in particular with those whose existence follows from the Equivariant Branching Lemma. It is demonstrated that symmetry can be exploited systematically to simplify computations of stabilities, and that in some cases symmetry constraints force instability of all bifurcating branches, found by using the Equivariant Branching Lemma, near the bifurcation point. Animal locomotion, which uses different time-periodic patterns having spatial regularity corresponding to permutations of the legs, is the main topic of Chapter 3, where classification of all possible spatio-temporal symmetry groups for time-periodic states of systems with a given spatial symmetry group is provided. Thorough analysis of the Hopf bifurcation, which is one of the most common routes to time periodicity, is carried out in Chapter 4. The proof of the Equivariant Hopf Theorem, an analogue of the Equivariant Branching Lemma, is sketched by using Lyapunov-Schmidt reduction from a suitable loop space. The closely related topic of Poincaré-Birkhoff normal form useful for studying stability of bifurcating branches is introduced. Equivariant Hopf bifurcation is illustrated with a discussion of spiral waves and target patterns in reaction-diffusion systems on a circular domain in the plane. Chapter 5 is concerned with lattice-symmetric equilibrium patterns in Euclidean-invariant PDEs. Although the Euclidean group is non-compact, assumption of lattice periodicity reduces the problem to one with a compact symmetry group. The authors examine six different applications with this kind of symmetry: reaction-diffusion systems, the Navier-Stokes equations, the primary visual cortex, liquid crystals, and Bénard convection with two different types of boundary conditions. Although each of these systems is symmetric under the Euclidean group in the plane, the systems exhibit different pattern-forming behavior, which leads to the conclusion that for an accurate analysis knowledge of the fact that the system is invariant under some symmetry group is not sufficient. In fact, information on at least three more issues related to existence of additional symmetries, appropriate representation of the symmetry group for a given problem, and interpretation of the symmetry in terms of structures that might be observed experimentally is required. Bifurcations from group orbits are studied in Chapter 6. The discussion is centered around the following property: when a solution loses stability and other solutions bifurcate, the stability is lost across the entire group orbit. Novel features of the bifurcation behavior in case when the group has a continuous subgroup and the orbits form manifolds are discussed. A detailed and complete account of the patterns which arise as a result of bifurcations in the Couette-Taylor flow is presented. Chapter 7 answers the question which symmetries of a given system should be taken into account through the discussion of hidden symmetries which appear in models posed on certain domains and with certain boundary conditions. In these cases the mathematical model can be embedded into a larger one with more symmetries, and the symmetries of a larger system determine the generic behavior of transitions in a smaller one. The effects of symmetry on heteroclinic cycles, used to explain several dynamic phenomena where a system repeatedly switches between different dynamical states, are examined in Chapter 8. It is shown, in particular, how heteroclinic cycles are related to the concept of bursting. Symmetric chaos is the subject of Chapter 9 aims at characterizing possible symmetry groups of chaotic attractors and developing several novel types of bifurcation that can occur in the presence of chaotic attractors. In particular, a phenomenon called bubbling which occurs, for instance, in the extensively studied problem of the synchronization of chaotic oscillators, is discussed. A practical experiment where the symmetry of a chaotic attractor can be observed is described. In contrast to the first nine chapters dealing with dissipative systems, the final Chapter 10 discusses application of the methods to Hamiltonian systems emphasizing the new features that arise because of their symplectic structure. The main result is an equivariant analogue of the Moser-Weinstein Theorem on the existence of families of periodic solutions near an equilibrium. Applications to many-body problems, the resonant spring pendulum, and molecular vibrations are considered. This excellent book reflects the authors' experience in exploration of the role of symmetry in explanation of pattern formation in nonlinear dynamics over the past fifteen years. The selection of the material, logical structure of the monograph and presentation are perfect. Numerous historical comments, carefully selected examples accompanied with helpful diagrams, graphs and illustrations, valuable bibliographic references combined with panoramic introductory pieces opening each chapter and rigorous mathematical exposition make this text an enjoyable reading for everyone interested in the role played by symmetries in nonlinear dynamics and equivariant bifurcation theory. Furthermore, the book can serve as a superb sample of mathematical writing.
scientific article; zbMATH DE number 1821147