List of chaotic maps

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
2D Lorenz system[1]	discrete	real	2	1	Euler method approximation to (non-chaotic) ODE.
2D Rational chaotic map[2]	discrete	rational	2	2
3-cells CNN system	continuous	real	3
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
Chialvo map	discrete	discrete		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] [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
Weierstrass map[125]	discrete	real	1	4	Gives rise to arbitrarily large global Lyapunov exponent
Willamowski-Rössler model[126]	continuous	real	3	10
WINDMI chaotic attractor[127][128][129]	continuous	real	1	2
Zaslavskii map	discrete	real	2	4
Zaslavskii rotation map
Zeraoulia-Sprott map[130]	discrete	real	2	2

List of fractals

See also: List of fractals by Hausdorff dimension

• 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