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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.
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 ( x n , y n ) in the plane and maps it to a new point given by
In dynamical systems theory, the Bogdanov map is a chaotic 2D map related to the Bogdanov–Takens bifurcation. It is given by the transformation: It is given by the transformation: { x n + 1 = x n + y n + 1 y n + 1 = y n + ϵ y n + k x n ( x n − 1 ) + μ x n y n {\displaystyle {\begin{cases}x_{n+1}=x_{n}+y_{n+1}\\y_{n+1}=y_{n}+\epsilon y_{n ...
The double-rod pendulum is one of the simplest dynamical systems with chaotic solutions. Chaos theory (or chaology [1]) is an interdisciplinary area of scientific study and branch of mathematics. It focuses on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions.
Depending on the value of μ, the tent map demonstrates a range of dynamical behaviour ranging from predictable to chaotic. If μ is less than 1 the point x = 0 is an attractive fixed point of the system for all initial values of x i.e. the system will converge towards x = 0 from any initial value of x.
In applied mathematics and astrodynamics, in the theory of dynamical systems, a crisis is the sudden appearance or disappearance of a strange attractor as the parameters of a dynamical system are varied. [1] [2] This global bifurcation occurs when a chaotic attractor comes into contact with an unstable periodic orbit or its stable manifold. [3]
A coupled map lattice (CML) is a dynamical system that models the behavior of nonlinear systems (especially partial differential equations).They are predominantly used to qualitatively study the chaotic dynamics of spatially extended systems.
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. The action of the map is defined geometrically by ...