1. Introduction: Background for Ordinary Differential Equations and Dynamical Systems.- 1.1. The Structure of Solutions of Ordinary Differential Equations.- 1.2. Conjugacies.- 1.3. Invariant Manifolds.- 1.4. Transversality, Structural Stability, and Genericity.- 1.5. Bifurcations.- 1.6. Poincaré Maps.- 2. Chaos: Its Descriptions and Conditions for Existence.- 2.1. The Smale Horseshoe.- 2.2. Symbolic Dynamics.- 2.3. Criteria for Chaos: The Hyperbolic Case.- 2.4. Criteria for Chaos: The Nonhyperbolic Case.- 3. Homoclinic and Heteroclinic Motions.- 3.1. Examples and Definitions.- 3.2. Orbits Homoclinic to Hyperbolic Fixed Points of Ordinary Differential Equations.- 3.3. Orbits Heteroclinic to Hyperbolic Fixed Points of Ordinary Differential Equations.- 3.4. Orbits Homoclinic to Periodic Orbits and Invariant Tori.- 4. Global Perturbation Methods for Detecting Chaotic Dynamics.- 4.1. The Three Basic Systems and Their Geometrical Structure.- 4.2. Examples.- 4.3. Final Remarks.- References.