Normally hyperbolic invariant manifolds in dynamical systems / Stephen Wiggins ; with the assistance of György Haller and Igor Mezic

This book is very attractive for anyone who wants to know how the invariant manifold theorems are applied in what areas of science and engineering. The author presents detailed proofs of three important invariant manifold theorems: (i) the persistence and smoothness theorem for overflowing invariant manifolds, (ii) the unstable manifold theorem for overflowing invariant manifolds, and (iii) the foliation theorem for unstable manifolds of overflowing invariant manifolds. The author first indicates briefly how these theorems can be applied in many areas in science and engineering such as fluid mechanics, theoretical chemistry, rigid body dynamics, chaotic scattering, condensed matter physics, mathematical biology, meteorology, plasma physics, celestial mechanics as well as chaos theory. The author next surveys some background in the mathematical development of the subject of invariant manifold theory such as the existence of invariant manifolds, the persistence and differentiability of invariant manifolds under perturbation and the behavior near an invariant manifold. The author also enumerates several methods for the invariant manifold theorems such as Lyapunov-Perron method, Hadamard method, Irvin’s method as well as the deformation method. After showing motivational examples the author gives some background material from differential geometry, beginning from the very definition of manifolds and tangent spaces. Next, an overflowing invariant manifold is defined. ... (zbMath)