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

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 - [1][2]
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]

[122]

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

See also: List of fractals by Hausdorff dimension

References

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  28. ^ Hopalong orbit fractal
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  125. ^ Internal fluctuations in a model of chemical chaos
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