The communication of knowledge on nonlinear dynamical systems, between the mathematicians working on the analytic approach and the scientists working mostly on the applications and numerical simulations has been less than ideal. This volume hopes to bridge the gap between books written on the subject by mathematicians and those written by scientists. The second objective of this volume is to draw attention to the need for cross-fertilization of knowledge between the physical and biological scientists. The third aim is to provide the reader with a personal guide on the study of global nonlinear dynamical systems.
Introduction; what is a dynamical system?; what is stability, and why should we care about it?. Part 1 Topics in topology and differential geometry: getting to the basics - algebra; bird's eye view of general topology; elementary differential topology and differential geometry; lie groups and group actions on manifolds; fibre bundles; vector bundles and tubular neighbourhood. Part 2 Introduction to global analysis and infinite dimensional manifolds: what is global analysis?; jet bundles; Whitney C topology; infinite dimensional manifolds; differential operators. Part 3 General theory of dynamical systems: equivalence relations; limiting sets and non-wandering sets; velocity fields, integrals, and ordinary differential equations; linear systems; linearization. Part 4 Stability theory by Liapunov's direct method: asymptotic stability and Liapunov's theorem; autonomous equations; Liapunov function; comparison method; how to construct Liapunov function. Part 5 Introduction to the general theory of structural stability: stable manifolds of diffeomorphisms and flows; low dimensional stable systems; anosov systems; structural stability; singularity of mappings; bifurcation. Part 6 Some established applications: feedback control; optical bistability and optical chaos; defects and dislocations in solids; fluid flow fields; reaction diffusion systems; stability in numerical analysis.