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In mathematics, a chaotic map is a map (= evolution function) that exhibits some sort of chaotic behavior. Maps may be parameterized by a discrete-time or a continuous-time parameter. Discrete maps usually take the form of iterated functions. Chaotic maps often occur in the study of dynamical systems.

Chaotic maps often generate fractals. Although a fractal may be constructed by an iterative procedure, some fractals are studied in and of themselves, as sets rather than in terms of the map that generates them. This is often because there are several different iterative procedures to generate the same fractal.

List of chaotic maps

List of fractals

• Cantor set
• de Rham curve
• Gravity set, or Mitchell-Green gravity set
• Julia set - derived from complex quadratic map
• Koch snowflake - special case of de Rham curve
• Lyapunov fractal
• Mandelbrot set - derived from complex quadratic map
• Menger sponge
• Newton fractal
• Nova fractal - derived from Newton fractal
• Quaternionic fractal - three dimensional complex quadratic map
• Sierpinski carpet
• Sierpinski triangle

v t e Chaos theory
Concepts	Core Attractor Bifurcation Fractal Limit set Lyapunov exponent Orbit Periodic point Phase space Anosov diffeomorphism Arnold tongue axiom A dynamical system Bifurcation diagram Box-counting dimension Correlation dimension Conservative system Ergodicity False nearest neighbors Hausdorff dimension Invariant measure Lyapunov stability Measure-preserving dynamical system Mixing Poincaré section Recurrence plot SRB measure Stable manifold Topological conjugacy Theorems Ergodic theorem Liouville's theorem Krylov–Bogolyubov theorem Poincaré–Bendixson theorem Poincaré recurrence theorem Stable manifold theorem Takens's theorem
Theoretical branches	Bifurcation theory Control of chaos Dynamical system Ergodic theory Quantum chaos Stability theory Synchronization of chaos
Chaotic maps (list)	Discrete Arnold's cat map Baker's map Complex quadratic map Coupled map lattice Duffing map Dyadic transformation Dynamical billiards outer Exponential map Gauss map Gingerbreadman map Hénon map Horseshoe map Ikeda map Interval exchange map Irrational rotation Kaplan–Yorke map Langton's ant Logistic map Standard map Tent map Tinkerbell map Zaslavskii map Continuous Double scroll attractor Duffing equation Lorenz system Lotka–Volterra equations Mackey–Glass equations Rabinovich–Fabrikant equations Rössler attractor Three-body problem Van der Pol oscillator
Physical systems	Chua's circuit Convection Double pendulum Elastic pendulum FPUT problem Hénon–Heiles system Kicked rotator Multiscroll attractor Population dynamics Swinging Atwood's machine Tilt-A-Whirl Weather
Chaos theorists	Michael Berry Rufus Bowen Mary Cartwright Chen Guanrong Leon O. Chua Mitchell Feigenbaum Peter Grassberger Celso Grebogi Martin Gutzwiller Brosl Hasslacher Michel Hénon Svetlana Jitomirskaya Bryna Kra Edward Norton Lorenz Aleksandr Lyapunov Benoît Mandelbrot Hee Oh Edward Ott Henri Poincaré Itamar Procaccia Mary Rees Otto Rössler David Ruelle Caroline Series Yakov Sinai Oleksandr Mykolayovych Sharkovsky Nina Snaith Floris Takens Audrey Terras Mary Tsingou Marcelo Viana Amie Wilkinson James A. Yorke Lai-Sang Young
Related articles	Butterfly effect Complexity Edge of chaos Predictability Santa Fe Institute

References

1. Chaos from Euler Solution of ODEs
2. Chaos from Euler Solution of ODEs
3. On the dynamics of a new simple 2-D rational discrete mapping
4. Chaos from Euler Solution of ODEs
5. http://www.yangsky.us/ijcc/pdf/ijcc83/IJCC823.pdf
6. The Aizawa attractor
7. Local Stability and Hopf Bifurcation Analysis of the Arneodo’s System
8. Basin of attraction Archived 2014-07-01 at the Wayback Machine
9. Image encryption based on new Beta chaotic maps
10. 1981 The Burke & Shaw system
11. A new chaotic attractor coined
12. A new chaotic attractor coined
13. http://www.scholarpedia.org/article/Chua_circuit Chua Circuit
14. Clifford Attractors
15. Peter de Jong Attractors
16. A discrete population model of delayed regulation
17. Irregular Attractors
18. A New Finance Chaotic Attractor
19. Hyperchaos Archived 2015-12-22 at the Wayback Machine
20. Visions of Chaos 2D Strange Attractor Tutorial
21. A new chaotic system and beyond: The generalized Lorenz-like system
22. Gingerbreadman map
23. Mira Fractals
24. Half-inverted tearing
25. Halvorsen: A tribute to Dr. Edward Norton Lorenz
26. Peter de Jong Attractors
27. Hopalong orbit fractal
28. Irregular Attractors
29. Global chaos synchronization of hyperchaotic chen system by sliding model control
30. Hyper-Lu system
31. The first hyperchaotic system
32. Hyperchaotic attractor Archived 2015-12-22 at the Wayback Machine
33. Attractors
34. Knot fractal map Archived 2015-12-22 at the Wayback Machine
36. A new discrete chaotic map based on the composition of permutations
37. A 3D symmetrical toroidal chaos
38. Lozi maps
39. Moore-Spiegel Attractor
40. A new chaotic system and beyond: The generalized lorenz-like system
41. A New Chaotic Jerk Circuit
42. Chaos Control and Hybrid Projective Synchronization of a Novel Chaotic System
43. Pickover
44. Polynomial Type-A
45. Polynomial Type-B
46. Polynomial Type-C
47. Quadrup Two Orbit Fractal
48. Rikitake chaotic attractor Archived 2010-06-20 at the Wayback Machine
49. Description of strange attractors using invariants of phase-plane
50. Skarya Archived 2015-12-22 at the Wayback Machine
51. Van der Pol Oscillator Equations
52. Shaw-Pol chaotic oscillator Archived 2015-12-22 at the Wayback Machine
53. The Shimiziu-Morioka System
54. Sprott B chaotic attractor Archived 2007-02-27 at the Wayback Machine
55. Chaos Blog - Sprott B system Archived 2015-12-22 at the Wayback Machine
56. Sprott C chaotic attractor Archived 2007-02-27 at the Wayback Machine
57. Chaos Blog - Sprott C system Archived 2015-12-22 at the Wayback Machine
58. Sprott's Gateway - Sprott-Linz A chaotic attractor Archived 2007-02-27 at the Wayback Machine
59. A new chaotic system and beyond: The generalized Lorenz-like System
60. Chaos Blog - Sprott-Linz A chaotic attractor Archived 2015-12-22 at the Wayback Machine
61. Sprott's Gateway - Sprott-Linz B chaotic attractor Archived 2007-02-27 at the Wayback Machine
62. A new chaotic system and beyond: The generalized Lorenz-like System
63. Chaos Blog - Sprott-Linz B chaotic attractor Archived 2015-12-22 at the Wayback Machine
64. Sprott's Gateway - Sprott-Linz C chaotic attractor Archived 2007-02-27 at the Wayback Machine
65. A new chaotic system and beyond: The generalized Lorenz-like System
66. Chaos Blog - Sprott-Linz C chaotic attractor Archived 2015-12-22 at the Wayback Machine
67. Sprott's Gateway - Sprott-Linz D chaotic attractor Archived 2007-02-27 at the Wayback Machine
68. A new chaotic system and beyond: The generalized Lorenz-like System
69. Chaos Blog - Sprott-Linz D chaotic attractor Archived 2015-12-22 at the Wayback Machine
70. Sprott's Gateway - Sprott-Linz E chaotic attractor Archived 2007-02-27 at the Wayback Machine
71. A new chaotic system and beyond: The generalized Lorenz-like System
72. Chaos Blog - Sprott-Linz E chaotic attractor Archived 2015-12-22 at the Wayback Machine
73. Sprott's Gateway - Sprott-Linz F chaotic attractor Archived 2007-02-27 at the Wayback Machine
74. A new chaotic system and beyond: The generalized Lorenz-like System
75. Chaos Blog - Sprott-Linz F chaotic attractor Archived 2015-12-22 at the Wayback Machine
76. Sprott's Gateway - Sprott-Linz G chaotic attractor Archived 2007-02-27 at the Wayback Machine
77. A new chaotic system and beyond: The generalized Lorenz-like System
78. Chaos Blog - Sprott-Linz G chaotic attractor Archived 2015-12-22 at the Wayback Machine
79. Sprott's Gateway - Sprott-Linz H chaotic attractor Archived 2007-02-27 at the Wayback Machine
80. A new chaotic system and beyond: The generalized Lorenz-like System
81. Chaos Blog - Sprott-Linz H chaotic attractor Archived 2015-12-22 at the Wayback Machine
82. Sprott's Gateway - Sprott-Linz I chaotic attractor Archived 2007-02-27 at the Wayback Machine
83. A new chaotic system and beyond: The generalized Lorenz-like System
84. Chaos Blog - Sprott-Linz I chaotic attractor Archived 2015-12-22 at the Wayback Machine
85. Sprott's Gateway - Sprott-Linz J chaotic attractor Archived 2007-02-27 at the Wayback Machine
86. A new chaotic system and beyond: The generalized Lorenz-like System
87. Chaos Blog - Sprott-Linz J chaotic attractor Archived 2015-12-22 at the Wayback Machine
88. Sprott's Gateway - Sprott-Linz K chaotic attractor Archived 2007-02-27 at the Wayback Machine
89. A new chaotic system and beyond: The generalized Lorenz-like System
90. Chaos Blog - Sprott-Linz K chaotic attractor Archived 2015-12-22 at the Wayback Machine
91. Sprott's Gateway - Sprott-Linz L chaotic attractor Archived 2007-02-27 at the Wayback Machine
92. A new chaotic system and beyond: The generalized Lorenz-like System
93. Chaos Blog - Sprott-Linz L chaotic attractor Archived 2015-12-22 at the Wayback Machine
94. Sprott's Gateway - Sprott-Linz M chaotic attractor Archived 2007-02-27 at the Wayback Machine
95. A new chaotic system and beyond: The generalized Lorenz-like System
96. Chaos Blog - Sprott-Linz M chaotic attractor Archived 2015-12-22 at the Wayback Machine
97. Sprott's Gateway - Sprott-Linz N chaotic attractor Archived 2007-02-27 at the Wayback Machine
98. A new chaotic system and beyond: The generalized Lorenz-like System
99. Chaos Blog - Sprott-Linz N chaotic attractor Archived 2015-12-22 at the Wayback Machine
100. Sprott's Gateway - Sprott-Linz O chaotic attractor Archived 2007-02-27 at the Wayback Machine
101. A new chaotic system and beyond: The generalized Lorenz-like System
102. Chaos Blog - Sprott-Linz O chaotic attractor Archived 2015-12-22 at the Wayback Machine
103. Sprott's Gateway - Sprott-Linz P chaotic attractor Archived 2007-02-27 at the Wayback Machine
104. A new chaotic system and beyond: The generalized Lorenz-like System
105. Chaos Blog - Sprott-Linz P chaotic attractor Archived 2015-12-22 at the Wayback Machine
106. Sprott's Gateway - Sprott-Linz Q chaotic attractor Archived 2007-02-27 at the Wayback Machine
107. A new chaotic system and beyond: The generalized Lorenz-like System
108. Chaos Blog - Sprott-Linz Q chaotic attractor Archived 2015-12-22 at the Wayback Machine
109. Sprott's Gateway - Sprott-Linz R chaotic attractor Archived 2007-02-27 at the Wayback Machine
110. A new chaotic system and beyond: The generalized Lorenz-like System
111. Chaos Blog - Sprott-Linz R chaotic attractor Archived 2015-12-22 at the Wayback Machine
112. Sprott's Gateway - Sprott-Linz S chaotic attractor Archived 2007-02-27 at the Wayback Machine
113. A new chaotic system and beyond: The generalized Lorenz-like System
114. Chaos Blog - Sprott-Linz S chaotic attractor Archived 2015-12-22 at the Wayback Machine
115. Strizhak-Kawczynski chaotic oscillator
116. Chaos Blog - Strizhak-Kawczynski chaotic oscillator Archived 2015-12-22 at the Wayback Machine
117. Sprott's Gateway - A symmetric chaotic flow
118. http://sprott.physics.wisc.edu/chaostsa/ Sprott's Gateway - Chaos and Time-Series Analysis
119. Oscillator of Ueda
120. Internal fluctuations in a model of chemical chaos
122. Synchronization of Chaotic Fractional-Order WINDMI Systems via Linear State Error Feedback Control
123. Adaptive Backstepping Controller Design for the Anti-Synchronization of Identical WINDMI Chaotic Systems with Unknown Parameters and its SPICE Implementation
124. Chen, Guanrong; Kudryashova, Elena V.; Kuznetsov, Nikolay V.; Leonov, Gennady A. (2016). "Dynamics of the Zeraoulia–Sprott Map Revisited". International Journal of Bifurcation and Chaos. 26 (7): 1650126–21. arXiv:1602.08632. Bibcode:2016IJBC...2650126C. doi:10.1142/S0218127416501261.

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