Mathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern as well as the classical techniques of applied mathematics. This renewal of interest, both in - search and teaching, has led to the establishment of the series Texts in Applied Mathematics (TAM). The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as nume- cal and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and to encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Mat- matical Sciences (AMS) series, whichwill focus on advanced textbooks and research-level monographs. Pasadena, California J.E. Marsden Providence, Rhode Island L. Sirovich College Park, Maryland S.S. Antman Preface to the Second Edition This edition contains a signi?cant amount of new material. The main r- son for this is that the subject of applied dynamical systems theory has seen explosive growth and expansion throughout the 1990s. Consequently, a student needs a much larger toolbox today in order to begin research on signi?cant problems.

This volume is intended for advanced undergraduate, graduates and researchers as an introduction to applied nonlinear dynamics and chaos. The author has placed emphasis on teaching the techniques and ideas which will enable students to take specific dynamical systems and obtain some quantitative information about the behaviour of these systems. The new edition has been updated and extended throughout and contains an extensive bibliography as well as a detailed glossary of terms.

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.