Lectures on Dynamical Systems, Structural Stability and Their Application

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.