This book was mostly written by a machine that was programmed to search a system of equations for chaotic solutions, simplify the equations to the extent possible, analyze the behavior, produce figures, and write the accompanying text. The equations are coupled autonomous ordinary differential equations with three variables and at least one nonlinearity. Fifty simple systems are included. Some are old and familiar; others are relatively new and unknown. They are chosen to illustrate by simple example most of dynamical behaviors that can occur in low-dimensional chaotic systems.There is no substitute for the thrill and insight of seeing the solution of a simple equation unfold as the trajectory wanders in real time across your computer screen using a program of your own making. A goal of this book is to inspire and delight as well as to teach. It provides a wealth of examples ripe for further study and extension, and it offers a glimpse of a future when artificial intelligence supplants many of the mundane tasks that accompany dynamical systems research and becomes a true and tireless collaborator.
Author(s): Julien C. Sprott
Year: 2023
Language: English
Pages: 348
Contents
Preface
1. Introduction
1.1 Background
1.2 Search Method
1.3 Simplified System
1.4 Equilibria
1.5 Attractor
1.6 Time Series
1.7 Lyapunov Exponents
1.8 Basin of Attraction
1.9 Bifurcations
1.10 Robustness
1.11 Conclusions
2. JCS-08-13-2022 System
2.1 Introduction
2.2 Simplified System
2.3 Equilibria
2.4 Attractor
2.5 Time Series
2.6 Lyapunov Exponents
2.7 Basin of Attraction
2.8 Bifurcations
2.9 Robustness
3. Lorenz System
3.1 Introduction
3.2 Simplified System
3.3 Equilibria
3.4 Attractor
3.5 Time Series
3.6 Lyapunov Exponents
3.7 Basin of Attraction
3.8 Bifurcations
3.9 Robustness
4. Rössler System
4.1 Introduction
4.2 Simplified System
4.3 Equilibria
4.4 Attractor
4.5 Time Series
4.6 Lyapunov Exponents
4.7 Basin of Attraction
4.8 Bifurcations
4.9 Robustness
5. Nosé–Hoover System
5.1 Introduction
5.2 Simplified System
5.3 Equilibria
5.4 Chaotic Sea
5.5 Time Series
5.6 Lyapunov Exponents
5.7 Extent of the Chaotic Sea
5.8 Bifurcations
5.9 Robustness
6. Diffusionless Lorenz System
6.1 Introduction
6.2 Simplified System
6.3 Equilibria
6.4 Attractor
6.5 Time Series
6.6 Lyapunov Exponents
6.7 Basin of Attraction
6.8 Bifurcations
6.9 Robustness
7. Sprott C System
7.1 Introduction
7.2 Simplified System
7.3 Equilibria
7.4 Attractor
7.5 Time Series
7.6 Lyapunov Exponents
7.7 Basin of Attraction
7.8 Bifurcations
7.9 Robustness
8. Sprott D System
8.1 Introduction
8.2 Simplified System
8.3 Equilibria
8.4 Attractor
8.5 Time Series
8.6 Lyapunov Exponents
8.7 Basin of Attraction
8.8 Bifurcations
8.9 Robustness
9. Sprott E System
9.1 Introduction
9.2 Simplified System
9.3 Equilibria
9.4 Attractor
9.5 Time Series
9.6 Lyapunov Exponents
9.7 Basin of Attraction
9.8 Bifurcations
9.9 Robustness
10. Sprott F System
10.1 Introduction
10.2 Simplified System
10.3 Equilibria
10.4 Attractor
10.5 Time Series
10.6 Lyapunov Exponents
10.7 Basin of Attraction
10.8 Bifurcations
10.9 Robustness
11. Sprott G System
11.1 Introduction
11.2 Simplified System
11.3 Equilibria
11.4 Attractor
11.5 Time Series
11.6 Lyapunov Exponents
11.7 Basin of Attraction
11.8 Bifurcations
11.9 Robustness
12. Sprott H System
12.1 Introduction
12.2 Simplified System
12.3 Equilibria
12.4 Attractor
12.5 Time Series
12.6 Lyapunov Exponents
12.7 Basin of Attraction
12.8 Bifurcations
12.9 Robustness
13. Sprott I System
13.1 Introduction
13.2 Simplified System
13.3 Equilibria
13.4 Attractor
13.5 Time Series
13.6 Lyapunov Exponents
13.7 Basin of Attraction
13.8 Bifurcations
13.9 Robustness
14. Sprott J System
14.1 Introduction
14.2 Simplified System
14.3 Equilibria
14.4 Attractor
14.5 Time Series
14.6 Lyapunov Exponents
14.7 Basin of Attraction
14.8 Bifurcations
14.9 Robustness
15. Sprott K System
15.1 Introduction
15.2 Simplified System
15.3 Equilibria
15.4 Attractor
15.5 Time Series
15.6 Lyapunov Exponents
15.7 Basin of Attraction
15.8 Bifurcations
15.9 Robustness
16. Sprott L System
16.1 Introduction
16.2 Simplified System
16.3 Equilibria
16.4 Attractor
16.5 Time Series
16.6 Lyapunov Exponents
16.7 Basin of Attraction
16.8 Bifurcations
16.9 Robustness
17. Sprott M System
17.1 Introduction
17.2 Simplified System
17.3 Equilibria
17.4 Attractor
17.5 Time Series
17.6 Lyapunov Exponents
17.7 Basin of Attraction
17.8 Bifurcations
17.9 Robustness
18. Sprott N System
18.1 Introduction
18.2 Simplified System
18.3 Equilibria
18.4 Attractor
18.5 Time Series
18.6 Lyapunov Exponents
18.7 Basin of Attraction
18.8 Bifurcations
18.9 Robustness
19. Sprott O System
19.1 Introduction
19.2 Simplified System
19.3 Equilibria
19.4 Attractor
19.5 Time Series
19.6 Lyapunov Exponents
19.7 Basin of Attraction
19.8 Bifurcations
19.9 Robustness
20. Sprott P System
20.1 Introduction
20.2 Simplified System
20.3 Equilibria
20.4 Attractor
20.5 Time Series
20.6 Lyapunov Exponents
20.7 Basin of Attraction
20.8 Bifurcations
20.9 Robustness
21. Sprott Q System
21.1 Introduction
21.2 Simplified System
21.3 Equilibria
21.4 Attractor
21.5 Time Series
21.6 Lyapunov Exponents
21.7 Basin of Attraction
21.8 Bifurcations
21.9 Robustness
22. Sprott R System
22.1 Introduction
22.2 Simplified System
22.3 Equilibria
22.4 Attractor
22.5 Time Series
22.6 Lyapunov Exponents
22.7 Basin of Attraction
22.8 Bifurcations
22.9 Robustness
23. Sprott S System
23.1 Introduction
23.2 Simplified System
23.3 Equilibria
23.4 Attractor
23.5 Time Series
23.6 Lyapunov Exponents
23.7 Basin of Attraction
23.8 Bifurcations
23.9 Robustness
24. Rössler Prototype-4 System
24.1 Introduction
24.2 Simplified System
24.3 Equilibria
24.4 Attractor
24.5 Time Series
24.6 Lyapunov Exponents
24.7 Basin of Attraction
24.8 Bifurcations
24.9 Robustness
25. Simplest Chaotic System
25.1 Introduction
25.2 Simplified System
25.3 Equilibria
25.4 Attractor
25.5 Time Series
25.6 Lyapunov Exponents
25.7 Basin of Attraction
25.8 Bifurcations
25.9 Robustness
26. Malasoma System
26.1 Introduction
26.2 Simplified System
26.3 Equilibria
26.4 Attractor
26.5 Time Series
26.6 Lyapunov Exponents
26.7 Basin of Attraction
26.8 Bifurcations
26.9 Robustness
27. Moore–Spiegel System
27.1 Introduction
27.2 Simplified System
27.3 Equilibria
27.4 Attractor
27.5 Time Series
27.6 Lyapunov Exponents
27.7 Basin of Attraction
27.8 Bifurcations
27.9 Robustness
28. Linz–Sprott System
28.1 Introduction
28.2 Simplified System
28.3 Equilibria
28.4 Attractor
28.5 Time Series
28.6 Lyapunov Exponents
28.7 Basin of Attraction
28.8 Bifurcations
28.9 Robustness
29. Elwakil–Kennedy System
29.1 Introduction
29.2 Simplified System
29.3 Equilibria
29.4 Attractor
29.5 Time Series
29.6 Lyapunov Exponents
29.7 Basin of Attraction
29.8 Bifurcations
29.9 Robustness
30. Chua System
30.1 Introduction
30.2 Simplified System
30.3 Equilibria
30.4 Attractor
30.5 Time Series
30.6 Lyapunov Exponents
30.7 Basin of Attraction
30.8 Bifurcations
30.9 Robustness
31. Chen System
31.1 Introduction
31.2 Simplified System
31.3 Equilibria
31.4 Attractor
31.5 Time Series
31.6 Lyapunov Exponents
31.7 Basin of Attraction
31.8 Bifurcations
31.9 Robustness
32. Halvorsen System
32.1 Introduction
32.2 Simplified System
32.3 Equilibria
32.4 Attractor
32.5 Time Series
32.6 Lyapunov Exponents
32.7 Basin of Attraction
32.8 Bifurcations
32.9 Robustness
33. Thomas System
33.1 Introduction
33.2 Simplified System
33.3 Equilibria
33.4 Attractor
33.5 Time Series
33.6 Lyapunov Exponents
33.7 Basin of Attraction
33.8 Bifurcations
33.9 Robustness
34. Rabinovich–Fabrikant System
34.1 Introduction
34.2 Simplified System
34.3 Equilibria
34.4 Attractor
34.5 Time Series
34.6 Lyapunov Exponents
34.7 Basin of Attraction
34.8 Bifurcations
34.9 Robustness
35. Leipnik–Newton System
35.1 Introduction
35.2 Simplified System
35.3 Equilibria
35.4 Attractor
35.5 Time Series
35.6 Lyapunov Exponents
35.7 Basin of Attraction
35.8 Bifurcations
35.9 Robustness
36. Arnéodo–Coullet–Tresser System
36.1 Introduction
36.2 Simplified System
36.3 Equilibria
36.4 Attractor
36.5 Time Series
36.6 Lyapunov Exponents
36.7 Basin of Attraction
36.8 Bifurcations
36.9 Robustness
37. Lorenz-84 System
37.1 Introduction
37.2 Simplified System
37.3 Equilibria
37.4 Attractor
37.5 Time Series
37.6 Lyapunov Exponents
37.7 Basin of Attraction
37.8 Bifurcations
37.9 Robustness
38. Wei System
38.1 Introduction
38.2 Simplified System
38.3 Equilibria
38.4 Attractor
38.5 Time Series
38.6 Lyapunov Exponents
38.7 Basin of Attraction
38.8 Bifurcations
38.9 Robustness
39. Wang–Chen System
39.1 Introduction
39.2 Simplified System
39.3 Equilibria
39.4 Attractor
39.5 Time Series
39.6 Lyapunov Exponents
39.7 Basin of Attraction
39.8 Bifurcations
39.9 Robustness
40. Reflection Symmetric System
40.1 Introduction
40.2 Simplified System
40.3 Equilibria
40.4 Attractor
40.5 Time Series
40.6 Lyapunov Exponents
40.7 Basin of Attraction
40.8 Bifurcations
40.9 Robustness
41. Buttery System
41.1 Introduction
41.2 Simplified System
41.3 Equilibria
41.4 Attractor
41.5 Time Series
41.6 Lyapunov Exponents
41.7 Basin of Attraction
41.8 Bifurcations
41.9 Robustness
42. Line Equilibrium System
42.1 Introduction
42.2 Simplified System
42.3 Equilibria
42.4 Attractor
42.5 Time Series
42.6 Lyapunov Exponents
42.7 Basin of Attraction
42.8 Bifurcations
42.9 Robustness
43. Mostly Quadratic System
43.1 Introduction
43.2 Simplified System
43.3 Equilibria
43.4 Attractor
43.5 Time Series
43.6 Lyapunov Exponents
43.7 Basin of Attraction
43.8 Bifurcations
43.9 Robustness
44. Dissipative–Conservative System
44.1 Introduction
44.2 Simplified System
44.3 Equilibria
44.4 Attractor
44.5 Time Series
44.6 Lyapunov Exponents
44.7 Basin of Attraction
44.8 Bifurcations
44.9 Robustness
45. Time-Reversible Reflection-Invariant System
45.1 Introduction
45.2 Simplified System
45.3 Equilibria
45.4 Attractor
45.5 Time Series
45.6 Lyapunov Exponents
45.7 Basin of Attraction
45.8 Bifurcations
45.9 Robustness
46. Plane Equilibrium System
46.1 Introduction
46.2 Simplified System
46.3 Equilibria
46.4 Attractor
46.5 Time Series
46.6 Lyapunov Exponents
46.7 Basin of Attraction
46.8 Bifurcations
46.9 Robustness
47. Forced Ueda System
47.1 Introduction
47.2 Simplified System
47.3 Equilibria
47.4 Attractor
47.5 Time Series
47.6 Lyapunov Exponents
47.7 Basin of Attraction
47.8 Bifurcations
47.9 Robustness
48. Megastable System
48.1 Introduction
48.2 Simplified System
48.3 Equilibria
48.4 Attractor
48.5 Time Series
48.6 Lyapunov Exponents
48.7 Basin of Attraction
48.8 Bifurcations
48.9 Robustness
49. Attracting Torus System
49.1 Introduction
49.2 Simplified System
49.3 Equilibria
49.4 Attractor
49.5 Time Series
49.6 Lyapunov Exponents
49.7 Basin of Attraction
49.8 Bifurcations
49.9 Robustness
50. Buncha System
50.1 Introduction
50.2 Simplified System
50.3 Equilibria
50.4 Attractor
50.5 Time Series
50.6 Lyapunov Exponents
50.7 Basin of Attraction
50.8 Bifurcations
50.9 Robustness
51. Signum Thermostat System
51.1 Introduction
51.2 Simplified System
51.3 Equilibria
51.4 Chaotic Sea
51.5 Time Series
51.6 Lyapunov Exponents
51.7 Extent of the Chaotic Sea
51.8 Bifurcations
51.9 Robustness
Bibliography
Index
About the Author
