Geometric Theory of Dynamical Systems: An Introduction

1 Differentiable Manifolds and Vector Fields.- §0 Calculus in ?n and Differentiable Manifolds.- §1 Vector Fields on Manifolds.- §2 The Topology of the Space of Cr Maps.- §3 Transversality.- §4 Structural Stability.- 2 Local Stability.- §1 The Tubular Flow Theorem.- §2 Linear Vector Fields.- §3 Singularities and Hyperbolic Fixed Points.- §4 Local Stability.- §5 Local Classification.- §6 Invariant Manifolds.- §7 The ?-lemma (Inclination Lemma). Geometrical Proof of Local Stability.- 3 The Kupka-Smale Theorem.- §1 The Poincaré Map.- §2 Genericity of Vector Fields Whose Closed Orbits Are Hyperbolic.- §3 Transversality of the Invariant Manifolds.- 4 Genericity and Stability of Morse-Smale Vector Fields.- §1 Morse-Smale Vector Fields; Structural Stability.- §2 Density of Morse-Smale Vector Fields on Orientable Surfaces.- §3 Generalizations.- §4 General Comments on Structural Stability. Other Topics.- References.