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. Language: English