Introduction to Applied Nonlinear Dynamical Systems and Chaos

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.

From the reviews of the second edition:"This is a very substantial revision of the author’s original textbook published in 1990. It does not only contain much new material, for instance on invariant manifold theory and normal forms, it has also been restructured. … The presentation is intended for advanced undergraduates … . This second edition … will serve as one of the most eminent introductions to the geometric theory of dynamical systems." (R. Bürger, Monatshefte für Mathematik, Vol. 145 (4), 2005)"This is an extensively rewritten version of the first edition which appeared in 1990, taking into account the many changes in the subject during the intervening time period. … The book is suitable for use as a textbook for graduate courses in applied mathematics or cognate fields. It is written in a readable style, with considerable motivation and many insightful examples. … Overall, the book provides a very accessible, up-to-date and comprehensive introduction to applied dynamical systems." (P.E. Kloeden, ZAMM-Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 85 (1), 2005)"The second edition of this popular text … is an encyclopedic introduction to dynamical systems theory and applications that includes substantial revisions and new material. It should be on the reading list of every student of the subject … . Also, the new organization makes the book more suitable as a textbook that can be used in graduate courses. This book will also be a useful reference for applied scientists … as well as a guide to the literature." (Carmen Chicone, Mathematical Reviews, 2004h)"This volume includes a significant amount of new material. … Each chapter starts with a narrative … and ends with a large collection of excellent exercises. … An extensive bibliography … provide a useful guide for future study. … This is a highly recommended book for advanced undergraduate and first-year graduate students. It contains most of the necessary mathematical tools … to apply the results of the subject to problems in the physical and engineering sciences." (Tibor Krisztin, Acta Scientiarum Mathematicarum, Vol. 75, 2009)“It is certainly one of the most complete introductory textbooks about dynamical systems, though no single book can be really complete. … Some chapters can certainly be used as a course text for a master’s course, but the whole book is to thick for a single course. … a suitable first text for Ph.D. students who want to do research in dynamical systems, and a useful reference work for more experienced people. I definitely enjoyed reading this book and can only recommend it.” (Kurt Lust, Bulletin of the Belgian Mathematical Society, Vol. 15 (1), 2008)