# Difference between revisions of "Strange Attractors"

Line 1: | Line 1: | ||

[[Image:Strange Attractor.jpg|thumb|350px|right|A visualization of the Poisson Saturne attractor, an example of a strange attractor.]] | [[Image:Strange Attractor.jpg|thumb|350px|right|A visualization of the Poisson Saturne attractor, an example of a strange attractor.]] | ||

==Basic Description== | ==Basic Description== | ||

− | A '''strange attractor''' is an infinite-point [[Field:Dynamic Systems#Jump3|attractor]] with [[Fractal Dimension|non-integer dimension]]. Although they consist of an infinite number of points, strange attractors do not fill [[Field:Dynamic Systems#Jump2|state space]]. Instead, they are contained within a bounded region and are often highly structured. In fact, Strange attractors are a type of [[Field:Fractals|fractal]], exhibiting self-similarity on all scales. | + | A '''strange attractor''', or '''chaotic attractor''', is an infinite-point [[Field:Dynamic Systems#Jump3|attractor]] with [[Fractal Dimension|non-integer dimension]]. Although they consist of an infinite number of points, strange attractors do not fill [[Field:Dynamic Systems#Jump2|state space]]. Instead, they are contained within a bounded region and are often highly structured. In fact, Strange attractors are a type of [[Field:Fractals|fractal]], exhibiting self-similarity on all scales. |

− | In [[Field:Dynamic Systems|dynamical systems theory]], strange attractors represent the dynamics of [[Chaos|chaotic]] systems. | + | In [[Field:Dynamic Systems|dynamical systems theory]], strange attractors represent the dynamics of [[Chaos|chaotic]] systems. |

==More Details== | ==More Details== | ||

If you examined the evolution of a system characterized by a strange attractor, you would notice some interesting things. As with any attractor, the trajectory of the system would migrate towards the strange attractor region of state space and return there if displaced. But a trajectory following a strange attractor would never repeat itself, no matter how long you watched. The system would never take on the exact same state twice. | If you examined the evolution of a system characterized by a strange attractor, you would notice some interesting things. As with any attractor, the trajectory of the system would migrate towards the strange attractor region of state space and return there if displaced. But a trajectory following a strange attractor would never repeat itself, no matter how long you watched. The system would never take on the exact same state twice. | ||

− | Furthermore, if you started the system at two similar states and watched the resulting evolution, you would see the two trajectories diverge from each other | + | Furthermore, if you started the system at two similar states and watched the resulting evolution, you would see the two trajectories diverge from each other [[Exponential Growth|exponentially]]. Even if the starting points were almost identical, given a little time, the resulting outcomes would look totally different from each other. This sensitivity to initial conditions is a hallmark of chaotic systems. |

+ | |||

+ | Strange attractors can be viewed as the set of all final states specified by chaotic solutions to the equations describing the system. | ||

+ | |||

+ | ==History== | ||

+ | The study of strange attractors began with the work of E.N. Lorenz and his 1963 paper "Deterministic non-periodic flow".<ref name=sourceone> Brin, M., & Stuck, G. (2002). ''Introduction to dynamical systems''. Cambridge; New York: Cambridge University Press.</ref> However, the term strange attractor was not used until the early 1970s when it was coined by David Ruelle and Floris Takens to describe turbulence.<ref name=sourcetwo> Sprott, J. (1993). ''Strange Attractors: creating patterns in chaos''. New York: Henry Holt & Company.</ref> | ||

==Examples of Strange Attractors== | ==Examples of Strange Attractors== | ||

Examples of strange attractors include the [[Henon Attractor| Hénon attractor]], [[Lorenz Attractor]], and [[Rössler attractor]]. | Examples of strange attractors include the [[Henon Attractor| Hénon attractor]], [[Lorenz Attractor]], and [[Rössler attractor]]. | ||

+ | |||

+ | ==References== | ||

+ | <references /> |

## Revision as of 09:56, 6 June 2012

## Basic Description

A **strange attractor**, or **chaotic attractor**, is an infinite-point attractor with non-integer dimension. Although they consist of an infinite number of points, strange attractors do not fill state space. Instead, they are contained within a bounded region and are often highly structured. In fact, Strange attractors are a type of fractal, exhibiting self-similarity on all scales.

In dynamical systems theory, strange attractors represent the dynamics of chaotic systems.

## More Details

If you examined the evolution of a system characterized by a strange attractor, you would notice some interesting things. As with any attractor, the trajectory of the system would migrate towards the strange attractor region of state space and return there if displaced. But a trajectory following a strange attractor would never repeat itself, no matter how long you watched. The system would never take on the exact same state twice.

Furthermore, if you started the system at two similar states and watched the resulting evolution, you would see the two trajectories diverge from each other exponentially. Even if the starting points were almost identical, given a little time, the resulting outcomes would look totally different from each other. This sensitivity to initial conditions is a hallmark of chaotic systems.

Strange attractors can be viewed as the set of all final states specified by chaotic solutions to the equations describing the system.

## History

The study of strange attractors began with the work of E.N. Lorenz and his 1963 paper "Deterministic non-periodic flow".^{[1]} However, the term strange attractor was not used until the early 1970s when it was coined by David Ruelle and Floris Takens to describe turbulence.^{[2]}

## Examples of Strange Attractors

Examples of strange attractors include the Hénon attractor, Lorenz Attractor, and Rössler attractor.