2-D quadratic maps and 3-D ODE systems : a rigorous approach /
This book is based on research on the rigorous proof of chaos and bifurcations in 2-D quadratic maps, especially the invertible case such as the Hňon map, and in 3-D ODE's, especially piecewise linear systems such as the Chua's circuit. In addition, the book covers some recent works in the...
Saved in:
| Main Author: | |
|---|---|
| Other Authors: | |
| Format: | Electronic eBook |
| Language: | English |
| Published: |
Singapore ; Hackensack, N.J. :
World Scientific,
©2010.
|
| Series: | World Scientific series on nonlinear science. Monographs and treatises ;
v. 73. |
| Subjects: | |
| Online Access: |
Full text (Emmanuel users only) |
Table of Contents:
- 1. Tools for the rigorous proof of chaos and bifurcations. 1.1. Introduction. 1.2. A chain of rigorous proof of chaos. 1.3. Poincare map technique. 1.4. The method of fixed point index. 1.5. Smale's horseshoe map. 1.6. The Sil'nikov criterion for the existence of chaos. 1.7. The Marotto theorem. 1.8. The verified optimization technique. 1.9. Shadowing lemma. 1.10. Method based on the second-derivative test and bounds for Lyapunov exponents. 1.11. The Wiener and Hammerstein cascade models. 1.12. Methods based on time series analysis. 1.13. A new chaos detector. 1.14. Exercises
- 2. 2-D quadratic maps : The invertible case. 2.1. Introduction. 2.2. Equivalences in the general 2-D quadratic maps. 2.3. Invertibility of the map. 2.4. The Henon map. 2.5. Methods for locating chaotic regions in the Henon map. 2.6. Bifurcation analysis. 2.7. Exercises
- 3. Classification of chaotic orbits of the general 2-D quadratic map. 3.1. Analytical prediction of system orbits. 3.2. A zone of possible chaotic orbits. 3.3. Boundary between different attractors. 3.4. Finding chaotic and nonchaotic attractors. 3.5. Finding hyperchaotic attractors. 3.6. Some criteria for finding chaotic orbits. 3.7. 2-D quadratic maps with one nonlinearity. 3.8. 2-D quadratic maps with two nonlinearities. 3.9. 2-D quadratic maps with three nonlinearities. 3.10. 2-D quadratic maps with four nonlinearities. 3.11. 2-D quadratic maps with five nonlinearities. 3.12. 2-D quadratic maps with six nonlinearities. 3.13. Numerical analysis
- 4. Rigorous proof of chaos in the double-scroll system. 4.1. Introduction. 4.2. Piecewise linear geometry and its real Jordan form. 4.3. The dynamics of an orbit in the double-scroll. 4.4. Poincare map [symbol]. 4.5. Method 1 : Sil'nikov criteria. 4.6. Subfamilies of the double-scroll family. 4.7. The geometric model. 4.8. Method 2 : The computer-assisted proof. 4.9. Exercises
- 5. Rigorous analysis of bifurcation phenomena. 5.1. Introduction. 5.2. Asymptotic stability of equilibria. 5.3. Types of chaotic attractors in the double-scroll. 5.4. Method 1 : Rigorous mathematical analysis. 5.5. Method 2 : One-dimensional Poincare map. 5.6. Exercises.