Progress in Mathematics

Sunyer 2001! Pattern formation in physical systems is one of the major research frontiers of mathematics. A central theme of The Symmetry Perspective is that many instances of pattern formation can be understood within a single framework: symmetry. The symmetries of a system of nonlinear ordinary or partial differential equations can be used to analyze, predict, and understand general mechanisms of pattern-formation. The symmetries of a system imply a 'catalogue' of typical forms of behavior, from which the actual behavior is 'selected'. A central theme is the distinction between phase space and physical space. The theory of nonlinear dynamical systems is largely discussed in terms of trajectories in an abstract phase space. The connection between the variables in the theory and the variables that are being observed can often be lost. Experimentalists typically observe a time series of measurements, whereas the abstract theory works with geometric objects such as attractors, homoclinic orbits, or invariant measures. Symmetries therefore provide an important route between the abstract theory and experimental observations. 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.