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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.
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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 ...
The main catalyst for the development of chaos theory was the electronic computer. Much of the mathematics of chaos theory involves the repeated iteration of simple mathematical formulas, which would be impractical to do by hand. Electronic computers made these repeated calculations practical, while figures and images made it possible to ...
It is a simple and pedagogical example for hyperbolic toral automorphisms. Thinking of the torus T 2 {\displaystyle \mathbb {T} ^{2}} as the quotient space R 2 / Z 2 {\displaystyle \mathbb {R} ^{2}/\mathbb {Z} ^{2}} , Arnold's cat map is the transformation Γ : T 2 → T 2 {\displaystyle \Gamma :\mathbb {T} ^{2}\to \mathbb {T} ^{2}} given by ...
Thus, the structure of the beam can be approximated by the kicked rotor. However, this map is interesting from a fundamental point of view in physics and mathematics because it is a very simple model of a conservative system that displays Hamiltonian chaos. It is therefore useful to study the development of chaos in this kind of system.
The dyadic transformation provides an example of how a simple 1-dimensional map can give rise to chaos. This map readily generalizes to several others. This map readily generalizes to several others. An important one is the beta transformation , defined as T β ( x ) = β x mod 1 {\displaystyle T_{\beta }(x)=\beta x{\bmod {1}}} .
In lab experiments that study chaos theory, approaches designed to control chaos are based on certain observed system behaviors. Any chaotic attractor contains an infinite number of unstable, periodic orbits. Chaotic dynamics, then, consists of a motion where the system state moves in the neighborhood of one of these orbits for a while, then ...