Introduction to Applied Nonlinear Dynamical Systems and Chaos This volume is intended for advanced undergraduate or first year graduate students as an introduction to applied nonlinear dynamics and chaos. The author has placed emphasis on teaching the techniques and ideas that will enable students to take specific dynamical systems and obtain some quantitative information about the behavior of these systems. He has included the basic core material that is necessary for higher levels of study and research. Thus people who do not necessarily have an extensive mathematical background such as students in engineering physics chemistry and biology will find this text as useful as students of mathematics. This new edition contains extensive new material on invariant manifold theory and normal forms (in particular Hamiltonian normal forms and the role of symmetry). Lagrangian Hamiltonian gradient and reversible dynamical systems are also discussed. Elementary Hamiltonian bifurcations are covered as well as the basic properties of circle maps. The book contains an extensive bibliography as well as a detailed glossary of terms making it a comprehensive book on applied nonlinear dynamical systems from a geometrical and analytical point of view. Equilibrium Solutions Stability and Linearized Stability Liapunov Functions Invariant Manifolds: Linear and Nonlinear Systems Periodic Orbits Vector Fields Possessing an Integral Index Theory Some General Properties of Vector Fields: Existence Uniqueness Differentiability and Flows Asymptotic Behavior The Poincaré Bendixson Theorem Poincaré Maps Conjugacies of Maps and Varying the Cross Section Structural Stability Genericity and Transversality Lagrange’s Equations Hamiltonian Vector Fields Gradient Vector Fields Reversible Dynamical Systems Asymptotically Autonomous Vector Fields Center Manifolds Normal Forms Bifurcation of Fixed Points of Vector Fields Bifurcations of Fixed Points of Maps On the Interpretation and Application of Bifurcation Diagrams: A Word of Caution The Smale Horseshoe Symbolic Dynamics The Conley Moser Conditions or “How to Prove That a Dynamical System is Chaotic” Dynamics Near Homoclinic Points of Two Dimensional Maps Orbits Homoclinic to Hyperbolic Fixed Points in Three Dimensional Autonomous Vector Fields Melnikov–s Method for Homoclinic Orbits in Two Dimensional Time Periodic Vector Fields Liapunov Exponents Chaos and Strange Attractors Hyperbolic Invariant Sets: A Chaotic Saddle Long Period Sinks in Dissipative Systems and Elliptic Islands in Conservative Systems Global Bifurcations Arising from Local Codimension—Two Bifurcations Glossary of Frequently Used Terms.