Synopses & Reviews
This book provides a self-contained comprehensive exposition of the theory of dynamical systems. The book begins with a discussion of several elementary but crucial examples. These are used to formulate a program for the general study of asymptotic properties and to introduce the principal theoretical concepts and methods. The main theme of the second part of the book is the interplay between local analysis near individual orbits and the global complexity of the orbit structure. The third and fourth parts develop the theories of low-dimensional dynamical systems and hyperbolic dynamical systems in depth. The book is aimed at students and researchers in mathematics at all levels from advanced undergraduate and up.
Synopsis
A self-contained comprehensive introduction to the mathematical theory of dynamical systems for students and researchers in mathematics, science and engineering.
Synopsis
A self-contained comprehensive introduction to the mathematical theory of dynamical systems for students and researchers in mathematics, science and engineering.
Synopsis
This book provides the first self-contained comprehensive exposition of the theory of dynamical systems as a core mathematical discipline closely intertwined with all main areas of mathematics. The authors introduce and rigorously develop the theory while providing researchers interested in applications with fundamental tools and paradigms. The book contains over 400 systematic exercises aimed at students and researchers in mathematics at all levels from advanced undergraduate up. Scientists and engineers working in applied dynamics, nonlinear science, and chaos will also find many fresh insights in this concrete and clear presentation.
Synopsis
Beginning with a discussion of several elementary but crucial examples, this study provides a self-contained comprehensive exposition of the theory of dynamical systems. It is aimed at students and researchers in mathematics at all levels from advanced undergraduate and up.
Table of Contents
Part I. Examples and Fundamental Concepts; Introduction; 1. First examples; 2. Equivalence, classification, and invariants; 3. Principle classes of asymptotic invariants; 4. Statistical behavior of the orbits and introduction to ergodic theory; 5. Smooth invariant measures and more examples; Part II. Local Analysis and Orbit Growth; 6. Local hyperbolic theory and its applications; 7. Transversality and genericity; 8. Orbit growth arising from topology; 9. Variational aspects of dynamics; Part III. Low-Dimensional Phenomena; 10. Introduction: What is low dimensional dynamics; 11. Homeomorphisms of the circle; 12. Circle diffeomorphisms; 13. Twist maps; 14. Flows on surfaces and related dynamical systems; 15. Continuous maps of the interval; 16. Smooth maps of the interval; Part IV. Hyperbolic Dynamical Systems; 17. Survey of examples; 18. Topological properties of hyperbolic sets; 19. Metric structure of hyperbolic sets; 20. Equilibrium states and smooth invariant measures; Part V. Sopplement and Appendix; 21. Dynamical systems with nonuniformly hyperbolic behavior Anatole Katok and Leonardo Mendoza.