List of chaotic maps
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In mathematics, a chaotic map is a map (an 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 and iterated functions often generate fractals. Some fractals are studied as objects themselves, as sets rather than in terms of the maps that generate them. This is often because there are several different iterative procedures that generate the same fractal. See also Universality (dynamical systems).
List of chaotic maps
Summarize
Perspective
| Map | Time domain | Space domain | Number of space dimensions | Number of parameters | Also known as |
|---|---|---|---|---|---|
| 3-cells CNN system | continuous | real | 3 | ||
| 2D Lorenz system[1] | discrete | real | 2 | 1 | Euler method approximation to (non-chaotic) ODE. |
| 2D Rational chaotic map[2] | discrete | rational | 2 | 2 | |
| ACT chaotic attractor [3] | continuous | real | 3 | ||
| Aizawa chaotic attractor[4] | continuous | real | 3 | 5 | |
| Arneodo chaotic system[5] | continuous | real | 3 | ||
| Arnold's cat map | discrete | real | 2 | 0 | |
| Baker's map | discrete | real | 2 | 0 | |
| Basin chaotic map[6] | discrete | real | 2 | 1 | |
| Beta Chaotic Map[7] | 12 | ||||
| Bogdanov map | discrete | real | 2 | 3 | |
| Brusselator | continuous | real | 3 | ||
| Burke-Shaw chaotic attractor[8] | continuous | real | 3 | 2 | |
| Chen chaotic attractor[9] | continuous | real | 3 | 3 | Not topologically conjugate to the Lorenz attractor. |
| Chen-Celikovsky system[10] | continuous | real | 3 | "Generalized Lorenz canonical form of chaotic systems" | |
| Chen-LU system[11] | continuous | real | 3 | 3 | Interpolates between Lorenz-like and Chen-like behavior. |
| Chen-Lee system | continuous | real | 3 | ||
| Chossat-Golubitsky symmetry map | |||||
| Chua circuit[12] | continuous | real | 3 | 3 | |
| Circle map | discrete | real | 1 | 2 | |
| Complex quadratic map | discrete | complex | 1 | 1 | gives rise to the Mandelbrot set |
| Complex squaring map | discrete | complex | 1 | 0 | acts on the Julia set for the squaring map. |
| Complex cubic map | discrete | complex | 1 | 2 | |
| Clifford fractal map[13] | discrete | real | 2 | 4 | |
| Degenerate Double Rotor map | |||||
| De Jong fractal map[14] | discrete | real | 2 | 4 | |
| Delayed-Logistic system[15] | discrete | real | 2 | 1 | |
| Discretized circular Van der Pol system[16] | discrete | real | 2 | 1 | Euler method approximation to 'circular' Van der Pol-like ODE. |
| Discretized Van der Pol system[17] | discrete | real | 2 | 2 | Euler method approximation to Van der Pol ODE. |
| Double rotor map | |||||
| Duffing map | discrete | real | 2 | 2 | Holmes chaotic map |
| Duffing equation | continuous | real | 2 | 5 (3 independent) | |
| Dyadic transformation | discrete | real | 1 | 0 | 2x mod 1 map, Bernoulli map, doubling map, sawtooth map |
| Exponential map | discrete | complex | 2 | 1 | |
| Feigenbaum strange nonchaotic map[18] | discrete | real | 3 | ||
| Finance system[19] | continuous | real | 3 | ||
| Folded-Towel hyperchaotic map[20] | continuous | real | 3 | ||
| Fractal-Dream system[21] | discrete | real | 2 | ||
| Gauss map | discrete | real | 1 | mouse map, Gaussian map | |
| Generalized Baker map | |||||
| Genesio-Tesi chaotic attractor[22] | continuous | real | 3 | ||
| Gingerbreadman map[23] | discrete | real | 2 | 0 | |
| Grinch dragon fractal | discrete | real | 2 | ||
| Gumowski/Mira map[24] | discrete | real | 2 | 1 | |
| Hadley chaotic circulation | continuous | real | 3 | 0 | |
| Half-inverted Rössler attractor[25] | |||||
| Halvorsen chaotic attractor[26] | continuous | real | 3 | ||
| Hénon map | discrete | real | 2 | 2 | |
| Hénon with 5th order polynomial | |||||
| Hindmarsh-Rose neuronal model | continuous | real | 3 | 8 | |
| Hitzl-Zele map | |||||
| Horseshoe map | discrete | real | 2 | 1 | |
| Hopa-Jong fractal[27] | discrete | real | 2 | ||
| Hopalong orbit fractal[28] | discrete | real | 2 | ||
| Hyper Logistic map[29] | discrete | real | 2 | ||
| Hyperchaotic Chen system[30] | continuous | real | 3 | ||
| Hyper Newton-Leipnik system[citation needed] | continuous | real | 4 | ||
| Hyper-Lorenz chaotic attractor | continuous | real | 4 | ||
| Hyper-Lu chaotic system[31] | continuous | real | 4 | ||
| Hyper-Rössler chaotic attractor[32] | continuous | real | 4 | ||
| Hyperchaotic attractor[33] | continuous | real | 4 | ||
| Ikeda chaotic attractor[34] | continuous | real | 3 | ||
| Ikeda map | discrete | real | 2 | 3 | Ikeda fractal map |
| Interval exchange map | discrete | real | 1 | variable | |
| Kaplan-Yorke map | discrete | real | 2 | 1 | |
| Knot fractal map[35] | discrete | real | 2 | ||
| Knot-Holder chaotic oscillator[36] | continuous | real | 3 | ||
| Kuramoto–Sivashinsky equation | continuous | real | |||
| Lambić map[37] | discrete | discrete | 1 | ||
| Li symmetrical toroidal chaos[38] | continuous | real | 3 | ||
| Linear map on unit square | |||||
| Logistic map | discrete | real | 1 | 1 | |
| Lorenz system | continuous | real | 3 | 3 | |
| Lorenz system's Poincaré return map | discrete | real | 2 | 3 | |
| Lorenz 96 model | continuous | real | arbitrary | 1 | |
| Lotka-Volterra system | continuous | real | 3 | 9 | |
| Lozi map[39] | discrete | real | 2 | ||
| Moore-Spiegel chaotic oscillator[40] | continuous | real | 3 | ||
| Scroll-Attractor[41] | continuous | real | 3 | ||
| Jerk Circuit[42] | continuous | real | 3 | ||
| Newton-Leipnik system | continuous | real | 3 | ||
| Nordmark truncated map | |||||
| Nosé-Hoover system | continuous | real | 3 | ||
| Novel chaotic system[43] | continuous | real | 3 | ||
| Pickover fractal map[44] | continuous | real | 3 | ||
| Pomeau-Manneville maps for intermittent chaos | discrete | real | 1 or 2 | Normal-form maps for intermittency (Types I, II and III) | |
| Polynom Type-A fractal map[45] | continuous | real | 3 | 3 | |
| Polynom Type-B fractal map[46] | continuous | real | 3 | 6 | |
| Polynom Type-C fractal map[47] | continuous | real | 3 | 18 | |
| Pulsed rotor | |||||
| Quadrup-Two orbit fractal[48] | discrete | real | 2 | 3 | |
| Quasiperiodicity map | |||||
| Mikhail Anatoly chaotic attractor | continuous | real | 3 | 2 | |
| Random Rotate map | |||||
| Rayleigh-Benard chaotic oscillator | continuous | real | 3 | 3 | |
| Rikitake chaotic attractor[49] | continuous | real | 3 | 3 | |
| Rössler attractor | continuous | real | 3 | 3 | |
| Rucklidge system[50] | continuous | real | 3 | 2 | |
| Sakarya chaotic attractor[51] | continuous | real | 3 | 2 | |
| Shaw-Pol chaotic oscillator[52][53] | continuous | real | 3 | 3 | |
| Shimizu-Morioka system[54] | continuous | real | 3 | 2 | |
| Shobu-Ose-Mori piecewise-linear map | discrete | real | 1 | piecewise-linear approximation for Pomeau-Manneville Type I map | |
| Sinai map - | |||||
| Sprott B chaotic system[55][56] | continuous | real | 3 | 2 | |
| Sprott C chaotic system[57][58] | continuous | real | 3 | 3 | |
| Sprott-Linz A chaotic attractor[59][60][61] | continuous | real | 3 | 0 | |
| Sprott-Linz B chaotic attractor[62][63][64] | continuous | real | 3 | 0 | |
| Sprott-Linz C chaotic attractor[65][66][67] | continuous | real | 3 | 0 | |
| Sprott-Linz D chaotic attractor[68][69][70] | continuous | real | 3 | 1 | |
| Sprott-Linz E chaotic attractor[71][72][73] | continuous | real | 3 | 1 | |
| Sprott-Linz F chaotic attractor[74][75][76] | continuous | real | 3 | 1 | |
| Sprott-Linz G chaotic attractor[77][78][79] | continuous | real | 3 | 1 | |
| Sprott-Linz H chaotic attractor[80][81][82] | continuous | real | 3 | 1 | |
| Sprott-Linz I chaotic attractor[83][84][85] | continuous | real | 3 | 1 | |
| Sprott-Linz J chaotic attractor[86][87][88] | continuous | real | 3 | 1 | |
| Sprott-Linz K chaotic attractor[89][90][91] | continuous | real | 3 | 1 | |
| Sprott-Linz L chaotic attractor[92][93][94] | continuous | real | 3 | 2 | |
| Sprott-Linz M chaotic attractor[95][96][97] | continuous | real | 3 | 1 | |
| Sprott-Linz N chaotic attractor[98][99][100] | continuous | real | 3 | 1 | |
| Sprott-Linz O chaotic attractor[101][102][103] | continuous | real | 3 | 1 | |
| Sprott-Linz P chaotic attractor[104][105][106] | continuous | real | 3 | 1 | |
| Sprott-Linz Q chaotic attractor[107][108][109] | continuous | real | 3 | 2 | |
| Sprott-Linz R chaotic attractor[110][111][112] | continuous | real | 3 | 2 | |
| Sprott-Linz S chaotic attractor[113][114][115] | continuous | real | 3 | 1 | |
| Standard map, Kicked rotor | discrete | real | 2 | 1 | Chirikov standard map, Chirikov-Taylor map |
| Strizhak-Kawczynski chaotic oscillator[116][117] | continuous | real | 3 | 9 | |
| Symmetric Flow attractor[118] | continuous | real | 3 | 1 | |
| Symplectic map | |||||
| Tangent map | |||||
| Tahn map[119] | discrete | real | 1 | 1 | Ring laser map [120]Beta distribution[121] |
| Thomas' cyclically symmetric attractor[123] | continuous | real | 3 | 1 | |
| Tent map | discrete | real | 1 | ||
| Tinkerbell map | discrete | real | 2 | 4 | |
| Triangle map | |||||
| Ueda chaotic oscillator[124] | continuous | real | 3 | 3 | |
| Van der Pol oscillator | continuous | real | 2 | 3 | |
| Willamowski-Rössler model[125] | continuous | real | 3 | 10 | |
| WINDMI chaotic attractor[126][127][128] | continuous | real | 1 | 2 | |
| Zaslavskii map | discrete | real | 2 | 4 | |
| Zaslavskii rotation map | |||||
| Zeraoulia-Sprott map[129] | discrete | real | 2 | 2 | |
| Chialvo map | discrete | discrete | 3 |
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
References
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