# Category: Chaotic maps

Artin billiard
In mathematics and physics, the Artin billiard is a type of a dynamical billiard first studied by Emil Artin in 1924. It describes the geodesic motion of a free particle on the non-compact Riemann sur
Hyperion (moon)
Hyperion /haɪˈpɪəriən/, also known as Saturn VII, is a moon of Saturn discovered by William Cranch Bond, his son George Phillips Bond and William Lassell in 1848. It is distinguished by its irregular
Logistic map
The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often referred to as an archetypal example of how complex, chaotic behaviour can arise from very simple non-li
Tent map
In mathematics, the tent map with parameter μ is the real-valued function fμ defined by the name being due to the tent-like shape of the graph of fμ. For the values of the parameter μ within 0 and 2,
Double pendulum
In physics and mathematics, in the area of dynamical systems, a double pendulum also known as a chaos pendulum is a pendulum with another pendulum attached to its end, forming a simple physical system
Hénon map
The Hénon map, sometimes called Hénon–Pomeau attractor/map, is a discrete-time dynamical system. It is one of the most studied examples of dynamical systems that exhibit chaotic behavior. The Hénon ma
Chialvo map
The Chialvo map is a two-dimensional map proposed by Dante R. Chialvo in 1995 to describe the generic dynamics of excitable systems. The model is inspired by Kunihiko Kaneko's Coupled map lattice (CML
Competitive Lotka–Volterra equations
The competitive Lotka–Volterra equations are a simple model of the population dynamics of species competing for some common resource. They can be further generalised to the Generalized Lotka–Volterra
Bogdanov map
In dynamical systems theory, the Bogdanov map is a chaotic 2D map related to the Bogdanov–Takens bifurcation. It is given by the transformation: The Bogdanov map is named after Rifkat Bogdanov.
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 b
Interval exchange transformation
In mathematics, an interval exchange transformation is a kind of dynamical system that generalises circle rotation. The phase space consists of the unit interval, and the transformation acts by cuttin
Tinkerbell map
The Tinkerbell map is a discrete-time dynamical system given by: Some commonly used values of a, b, c, and d are * * Like all chaotic maps, the Tinkerbell Map has also been shown to have periods; af
List of chaotic maps
In mathematics, a chaotic map is a map (namely, an evolution function) that exhibits some sort of chaotic behavior. Maps may be parameterized by a discrete-time or a continuous-time parameter. Discret
Rabinovich–Fabrikant equations
The Rabinovich–Fabrikant equations are a set of three coupled ordinary differential equations exhibiting chaotic behaviour for certain values of the parameters. They are named after Mikhail Rabinovich
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. It can be constructed by performing the following
Brusselator
The Brusselator is a theoretical model for a type of autocatalytic reaction.The Brusselator model was proposed by Ilya Prigogine and his collaborators at the Université Libre de Bruxelles. It is a por
Mackey–Glass equations
In mathematics and mathematical biology, the Mackey–Glass equations, named after Michael Mackey and Leon Glass, refer to a family of delay differential equations whose behaviour manages to mimic both
Rössler attractor
The Rössler attractor /ˈrɒslər/ is the attractor for the Rössler system, a system of three non-linear ordinary differential equations originally studied by Otto Rössler in the 1970s. These differentia
Elastic pendulum
In physics and mathematics, in the area of dynamical systems, an elastic pendulum (also called spring pendulum or swinging spring) is a physical system where a piece of mass is connected to a spring s
Chua's circuit
Chua's circuit (also known as a Chua circuit) is a simple electronic circuit that exhibits classic chaotic behavior. This means roughly that it is a "nonperiodic oscillator"; it produces an oscillatin
Kuramoto–Sivashinsky equation
In mathematics, the Kuramoto–Sivashinsky equation (also called the KS equation or flame equation) is a fourth-order nonlinear partial differential equation. It is named after Yoshiki Kuramoto and Greg
Ikeda map
In physics and mathematics, the Ikeda map is a discrete-time dynamical system given by the complex map The original map was proposed first by as a model of light going around across a nonlinear optica
Lorenz 96 model
The Lorenz 96 model is a dynamical system formulated by Edward Lorenz in 1996. It is defined as follows. For : where it is assumed that and and . Here is the state of the system and is a forcing const
Colpitts oscillator
A Colpitts oscillator, invented in 1918 by American engineer Edwin H. Colpitts, is one of a number of designs for LC oscillators, electronic oscillators that use a combination of inductors (L) and cap
Mixmaster universe
The Mixmaster universe (named after Sunbeam Mixmaster, a brand of Sunbeam Products electric kitchen mixer) is a solution to Einstein field equations of general relativity studied by Charles Misner in
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
Three-body problem
In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion acco
The dyadic transformation (also known as the dyadic map, bit shift map, 2x mod 1 map, Bernoulli map, doubling map or sawtooth map) is the mapping (i.e., recurrence relation) (where is the set of seque
Kicked rotator
The kicked rotator, also spelled as kicked rotor, is a paradigmatic model for both Hamiltonian chaos (the study of chaos in Hamiltonian systems) and quantum chaos. It describes a free rotating stick (
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 (xn, yn ) in the plane and maps it t
Lorenz system
The Lorenz system is a system of ordinary differential equations first studied by mathematician and meteorologist Edward Lorenz. It is notable for having chaotic solutions for certain parameter values
Duffing map
The Duffing map (also called as 'Holmes map') is a discrete-time dynamical system. It is an example of a dynamical system that exhibits chaotic behavior. The Duffing map takes a point (xn, yn) in the
Van der Pol oscillator
In dynamics, the Van der Pol oscillator is a non-conservative oscillator with non-linear damping. It evolves in time according to the second-order differential equation: where x is the position coordi
Thomas' cyclically symmetric attractor
In the dynamical systems theory, Thomas' cyclically symmetric attractor is a 3D strange attractor originally proposed by René Thomas. It has a simple form which is cyclically symmetric in the x,y, and
Coupled map lattice
A coupled map lattice (CML) is a dynamical system that models the behavior of non-linear systems (especially partial differential equations). They are predominantly used to qualitatively study the cha
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. Thinking o
Bak–Sneppen model
The Bak–Sneppen model is a simple model of co-evolution between interacting species. It was developed to show how self-organized criticality may explain key features of the fossil record, such as the
Standard map
The standard map (also known as the Chirikov–Taylor map or as the Chirikov standard map) is an area-preserving chaotic map from a square with side onto itself. It is constructed by a Poincaré's surfac
Arnold tongue
In mathematics, particularly in dynamical systems, Arnold tongues (named after Vladimir Arnold) are a pictorial phenomenon that occur when visualizing how the rotation number of a dynamical system, or
Chirikov criterion
The Chirikov criterion or Chirikov resonance-overlap criterionwas established by the Russian physicist Boris Chirikov.Back in 1959, he published a seminal article,where he introduced the very first ph
Hindmarsh–Rose model
The Hindmarsh–Rose model of neuronal activity is aimed to study the spiking-bursting behavior of the membrane potential observed in experiments made with a single neuron. The relevant variable is the
Duffing equation
The Duffing equation (or Duffing oscillator), named after (1861–1944), is a non-linear second-order differential equation used to model certain damped and driven oscillators. The equation is given by