Chaos in Discrete Dynamical Systems: A Visual Introduction in 2 Dimensions

by Ralph H. Abraham

of the Book.- 1 Introduction.- 2 Basic concepts in 1D.- 3 Basic concepts in 2D.- 4 Absorbing Areas.- 5 Holes.- 6 Fractal Boundaries.- 7 Chaotic Contact Bifurcations.- 8 Conclusion.- Appendix 1 Notations.- A1.1 Formal logic.- A1.2 Set theory.- A1.3 Point set topology.- Appendix 2 Topological Dynamics.- A2.1 Trajectories and orbits.- A2.2 Inverse images.- A2.3 Fixed points.- A2.4 Periodic trajectories.- A2.5 Limit points.- A2.6 Stable sets, attractors, and basins.- A2.7 Unstable sets and repellors.- A2.8 Chaotic attractors.- Appendix 3 Critical Curves.- A3.1 The zones.- A3.2 Critical points via calculus.- A3.3 Critical points via topology.- A3.4 The critical curves.- A3.5 Absorbing areas.- Appendix 4 Synonyms.- Appendix 5 History, Part 1.- A5.1 Early history.- A5.2 Finite difference equations.- A5.3 Functional equations.- A5.4 Poincaré.- A5.5 Independent contemporaries of Poincaré.- A5.6 Birkhoff.- A5.7 Denjoy.- A5.8 The Russian school.- A5.9 The Japanese school.- A5.10 Conservative systems.- A5.11 The American school.- A5.12 Numerical methods and applied work.- A5.13 Iteration theory.- A5.14 The methods of Liapunov.- A5.15 Periodic solutions.- A5.16 Control theory.- A5.17 Other applications.- A5.18 Conclusion.- A5.19 Historical Bibliography.- Appendix 6 History, Part 2.- A6.1 Introduction.- A6.2 G.D. Birkhoff.- A6.3 Nonlinear oscillations from 1925.- A6.4 The Mandelstham-Andronov school.- A6.5 The Bogoliubov (or Kiev) school.- A6.6 Poincaré's analyticity theorem.- A6.7 Myrberg's contribution.- A6.8 Conclusion.- Appendix 7 Domains of the Figures.- Frequently used references by code.- Bibliography by author.

Genres: Science

282 pages, Paperback

First published May 29, 1997

About the author

Dr. Ralph Herman Abraham was a mathematician who specialized in the development of dynamical systems theory. He was a Professor of Mathematics at the University of California, Santa Cruz. He also consulted on chaos theory and its applications in fields such as medical physiology, ecology, mathematical economics, and psychotherapy.

Abraham was also interested in alternative ways of expressing mathematics, for example visually or aurally. He has staged performances in which mathematics, visual arts and music are combined into one presentation. He also developed an interest in "Hip" activities in Santa Cruz in the 1960s and he wrote several books and one website on the topic. He credited his use of psychedelics for inspiring this interest.