Abstract
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.
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