The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space (Progress in Mathematics, 200)

Pattern formation in physical systems is one of the major research frontiers of mathematics. A central theme of this book is that many instances of pattern formation can be understood within a single framework: symmetry. The book applies symmetry methods to increasingly complex kinds of dynamic behavior: equilibria, period-doubling, time-periodic states, homoclinic and heteroclinic orbits, and chaos. Examples are drawn from both ODEs and PDEs. In each case the type of dynamical behavior being studied is motivated through applications, drawn from a wide variety of scientific disciplines ranging from theoretical physics to evolutionary biology. An extensive bibliography is provided.

pattern Formation In Physical Systems Is One Of The Major Research Frontiers Of Mathematics. A Central Theme Of This Book Is That Many Instances Of Pattern Formation Can Be Understood Within A Single Framework: Symmetry. This Book Applies Symmetry Methods To Increasingly Complex Kinds Of Dynamic Behavior: Equilibria, Period-doubling, Time-periodic States, Homoclinic And Heteroclinic Orbits, And Chaos. Examples Are Drawn From Both Odes And Pdes. In Each Case The Type Of Dynamical Behavior Being Studied Is Motivated Through Applications, Drawn From A Wide Variety Of Scientific Disciplines Ranging From Theoretical Physics To Evolutionary Biology.

Front Matter....Pages i-xvii Steady-State Bifurcation....Pages 1-31 Linear Stability....Pages 33-57 Time Periodicity and Spatio-Temporal Symmetry....Pages 59-86 Hopf Bifurcation with Symmetry....Pages 87-122 Steady-State Bifurcations in Euclidean Equivariant Systems....Pages 123-159 Bifurcation From Group Orbits....Pages 161-199 Hidden Symmetry and Genericity....Pages 201-220 Heteroclinic Cycles....Pages 221-240 Symmetric Chaos....Pages 241-274 Periodic Solutions of Symmetric Hamiltonian Systems....Pages 275-300 Back Matter....Pages 301-325 The framework of symmetry provides an important route between the abstract theory and experimental observations. The book applies symmetry methods to dynamical systems, focusing on bifurcation and chaos theory. Its exposition is organized around a wide variety of relevant applications. From the reviews: -- "[The] rich collection of examples makes the book...extremely useful for motivation and for spreading the ideas to a large Community."--MATHEMATICAL REVIEWS Most research in dynamical systems theory is currently focused on exotic behavior such as chaos.