In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a chaotic map is a map

Map (mathematics)

In most of mathematics and in some related technical fields, the term mapping, usually shortened to map, is either a synonym for function, or denotes a particular kind of function which is important in that branch, or denotes something conceptually similar to a function.In graph theory, a map is a...

 (= 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 function

Iterated function

In mathematics, an iterated function is a function which is composed with itself, possibly ad infinitum, in a process called iteration. In this process, starting from some initial value, the result of applying a given function is fed again in the function as input, and this process is repeated...

s. Chaotic maps often occur in the study of dynamical system

Dynamical system

A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...

s.

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

Map	Time domain	Space domain	Number of space dimensions	Also known as
Arnold's cat map Arnold's cat map In mathematics, Arnold's cat map is a chaotic map from the torus into itself, named after Vladimir Arnold, who demonstrated its effects in the 1960s using an image of a cat, hence the name....	discrete	real	2
Baker's map Baker's map In dynamical systems theory, the baker's map is a chaotic map from the unit square into itself. It is named after a kneading operation that bakers apply to dough: the dough is cut in half, and the two halves are stacked on one-another, and compressed...	discrete	real	2
Bogdanov map
Chossat-Golubitsky symmetry map
Circle map Circle map In mathematics, a circle map is a member of a family of dynamical systems on the circle first defined by Andrey Kolmogorov. Kolmogorov proposed this family as a simplified model for driven mechanical rotors . The circle map equations also describe a simplified model of the phase-locked loop in...	discrete	real	1
Complex quadratic map	discrete	complex	1
Complex squaring map Complex squaring map In mathematics, the complex squaring map, a polynomial mapping of degree two, is a simple and accessible demonstration of chaos in dynamical systems...	discrete	complex	1
Complex Cubic map
Degenerate Double Rotor map
Double Rotor map
Duffing map	discrete	real	2
Duffing equation	continuous	real	1
Dyadic transformation Dyadic transformation The dyadic transformation is the mapping produced by the rule...	discrete	real	1	2x mod 1 map, Bernoulli map, doubling map, sawtooth map
Exponential map Exponential map (discrete dynamical systems) In the theory of dynamical systems, the exponential map can be used as the evolution function of the discrete nonlinear dynamical system.The family of exponential functions is called exponential family....	discrete	complex	2
Gauss map Gauss iterated map In mathematics, the Gauss map , is a nonlinear iterated map of the reals into a real interval given by the Gaussian function:where α and β are real parameters....	discrete	real	1	mouse map, Gaussian map
Generalized Baker map
Gingerbreadman map	discrete	real	2
Gumowski/Mira map
Hénon map	discrete	real	2
Hénon with 5th order polynomial
Hitzl-Zele map
Horseshoe map Horseshoe map In the mathematics of chaos theory, a horseshoe map is any member of a class of chaotic maps of the square into itself. It is a core example in the study of dynamical systems. The map was introduced by Stephen Smale while studying the behavior of the orbits of the van der Pol oscillator...	discrete	real	2
Ikeda map	discrete	real	2
Interval exchange map	discrete	real	1
Kaplan-Yorke map Kaplan-Yorke map The Kaplan–Yorke map is a discrete-time dynamical system. It is an example of a dynamical system that exhibits chaotic behavior. The Kaplan–Yorke map takes a point in the plane and maps it to a new point given by...	discrete	real	2
Linear map on unit square
Logistic map Logistic map The logistic map is a polynomial mapping of degree 2, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations...	discrete	real	1
Lorenz attractor Lorenz attractor The Lorenz attractor, named for Edward N. Lorenz, is an example of a non-linear dynamic system corresponding to the long-term behavior of the Lorenz oscillator. The Lorenz oscillator is a 3-dimensional dynamical system that exhibits chaotic flow, noted for its lemniscate shape...	continuous	real	3
Lorenz system's Poincare Return map
Lozi map	discrete	real	2
Nordmark truncated map
Pomeau-Manneville maps for intermittent chaos	discrete	real	1 and 2	Normal-form maps for intermittency (Types I, II and III)
Pulsed rotor
Quasiperiodicity map
Rabinovich-Fabrikant equations Rabinovich-Fabrikant equations The Rabinovich–Fabrikant equations are a set of three coupled ordinary differential equations exhibiting chaotic behavior for certain values of the parameters...	continuous	real	3
Random Rotate map
Rössler map	continuous	real	3
Shobu-Ose-Mori piecewise-linear map	discrete	real	1	piecewise-linear approximation for Pomeau-Manneville Type I map
Sinai map - See http://www.maths.ox.ac.uk/~mcsharry/papers/dynsys18n3p191y2003mcsharry.pdf
Symplectic map
Standard map, Kicked rotor	discrete	real	2	Chirikov standard map, Chirikov-Taylor map
Tangent map
Tent map	discrete	real	1
Tinkerbell map	discrete	real	2
Triangle map
Van der Pol oscillator	continuous	real	1
Zaslavskii map Zaslavskii map The Zaslavskii map is a discrete-time dynamical system. It is an example of a dynamical system that exhibits chaotic behavior. The Zaslavskii map takes a point in the plane and maps it to a new point:...	discrete	real	2
Zaslavskii rotation map

List of fractals

• Cantor set
  Cantor set
  In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties. It was discovered in 1875 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883....
  
  
• de Rham curve
  De Rham curve
  In mathematics, a de Rham curve is a certain type of fractal curve named in honor of Georges de Rham.The Cantor function, Césaro curve, Minkowski's question mark function, the Lévy C curve, the blancmange curve and the Koch curve are all special cases of the general de Rham...
  
  
• Gravity set, or Mitchell-Green gravity set
• Julia set
  Julia set
  In the context of complex dynamics, a topic of mathematics, the Julia set and the Fatou set are two complementary sets defined from a function...
  
   - derived from complex quadratic map
• Newton fractal
• Nova fractal
  Nova fractal
  Nova fractal is a family of fractals related to the Newton fractal. Nova is a formula that is implemented in most fractal art software.-Formula:The formula for novaMandelbrot is a special case of the generalized Newton fractal:...
  
   - derived from Newton fractal
• Koch snowflake
  Koch snowflake
  The Koch snowflake is a mathematical curve and one of the earliest fractal curves to have been described...
  
   - special case of de Rham curve
  De Rham curve
  In mathematics, a de Rham curve is a certain type of fractal curve named in honor of Georges de Rham.The Cantor function, Césaro curve, Minkowski's question mark function, the Lévy C curve, the blancmange curve and the Koch curve are all special cases of the general de Rham...
  
  
• Lyapunov fractal
  Lyapunov fractal
  In mathematics, Lyapunov fractals are bifurcational fractals derived from an extension of the logistic map in which the degree of the growth of the population, r, periodically switches between two values A and B.A Lyapunov fractal is constructed by mapping the regions of stability and chaotic...
  
  
• Mandelbrot set
  Mandelbrot set
  The Mandelbrot set is a particular mathematical set of points, whose boundary generates a distinctive and easily recognisable two-dimensional fractal shape...
  
    - derived from complex quadratic map
• Menger sponge
  Menger sponge
  In mathematics, the Menger sponge is a fractal curve. It is a universal curve, in that it has topological dimension one, and any other curve is homeomorphic to some subset of it. It is sometimes called the Menger-Sierpinski sponge or the Sierpinski sponge...
  
  
• Sierpinski carpet
  Sierpinski carpet
  The Sierpinski carpet is a plane fractal first described by Wacław Sierpiński in 1916. The carpet is a generalization of the Cantor set to two dimensions . Sierpiński demonstrated that this fractal is a universal curve, in that any possible one-dimensional graph, projected onto the two-dimensional...
  
  
• Sierpinski triangle
  Sierpinski triangle
  The Sierpinski triangle , also called the Sierpinski gasket or the Sierpinski Sieve, is a fractal and attractive fixed set named after the Polish mathematician Wacław Sierpiński who described it in 1915. However, similar patterns appear already in the 13th-century Cosmati mosaics in the cathedral...