If the canonical treatment of perfectly regular flows reduces them to circular motions in action-angle space, what geometry should we expect for completely chaotic flows? If integrable quantum systems have spectra which can be expressed using simple algebraic expressions, what should we expect for quantum systems with no order? In this text, Cvitanović and colleagues explore this rich vein of theory. The point of view is that if complete order and complete chaos are two limiting behaviours, the latter should have as rigorous and complete a mathematical description as the former.
The discussion starts with identifying equilibiria and periodic orbits as the important invariant structures upon which to build a theory. A grammar and language is developed for these structures. Through a variety of examples, it is shown how important system properties such as escape rates, stability exponents, diffusion constants can be defined and calculated. Equivalently in the quantum world, quantum spectra and scattering rates can be determined.
The authors lead the reader in exploring a wide variety of related topics including group theory and discrete symmetries, turbulence and infinite dimensional flows, Hamiltonian and dissipative systems, WKB and other semi-classical approximations, among many others. The coverage aims not to overwhelm with formalism while being as self-contained as practical. The book is the culmination of more than two decades of cutting edge research by some of the most talented theorists in the area of dynamical systems. The authors' love of the topic shows in the lively and engaging text, as does their deep esthetic appreciation for the geometry and algebra of chaotic systems.
Unlike most expository texts, there is a rich set of exercises making this text useful both as a reference on the topic and a self-contained text for a graduate-level course.