Difference between revisions of "Strange Attractors"

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[[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''', 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.   
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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 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]], the dynamics of [[Chaos|chaotic]] systems are represented by strange attractors.   
 
In [[Field:Dynamic Systems|dynamical systems theory]], the dynamics of [[Chaos|chaotic]] systems are represented by strange attractors.   
  
 
==Strange Trajectories==
 
==Strange Trajectories==
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 eventually migrate to 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.  This is because systems described by strange attractors are <balloon title="A periodic system has repeating cycles, which allows us to make predictions about what it will do in the future.  Non-periodic systems have no predictable cycles.">non-periodic</balloon>
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If you examined the evolution of a system characterized by a strange attractor, you would notice some interesting things. As with any attractor, nearby trajectories of the system would migrate to 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.  This is because systems described by strange attractors are <balloon title="A periodic system has repeating cycles, which allows us to make predictions about what it will do in the future.  Non-periodic systems have no predictable cycles.">non-periodic</balloon>
  
 
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.  
 
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 Structure==
 
==Strange Structure==
Strange attractors are generated by graphing or plotting equations in phase space. The equations describing a strange attractor can be differential equations, as in the case of the [[Lorenz Attractor]], or difference equations, as in the case of the [[Henon Attractor| Hénon Attractor]].
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Strange attractors are generated by graphing or plotting certain [[nonlinearity|nonlinear]] equations in phase space. The equations describing a strange attractor can be differential equations, as in the case of the [[Lorenz Attractor]], or difference equations, as in the case of the [[Henon Attractor| Hénon Attractor]].
  
 
Strange attractors can be viewed as the set of all final states specified by chaotic solutions to the system's equations.  Depending on the system, all solutions or only a subset of all the solutions could yield a strange attractor.
 
Strange attractors can be viewed as the set of all final states specified by chaotic solutions to the system's equations.  Depending on the system, all solutions or only a subset of all the solutions could yield a strange attractor.

Revision as of 09:01, 7 June 2012

A visualization of the Poisson Saturne attractor, an example of a strange attractor.

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 highly structured. In fact, Strange attractors are a type of fractal, exhibiting self-similarity on all scales.

In dynamical systems theory, the dynamics of chaotic systems are represented by strange attractors.

Strange Trajectories

If you examined the evolution of a system characterized by a strange attractor, you would notice some interesting things. As with any attractor, nearby trajectories of the system would migrate to 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. This is because systems described by strange attractors are non-periodic

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 Structure

Strange attractors are generated by graphing or plotting certain nonlinear equations in phase space. The equations describing a strange attractor can be differential equations, as in the case of the Lorenz Attractor, or difference equations, as in the case of the Hénon Attractor.

Strange attractors can be viewed as the set of all final states specified by chaotic solutions to the system's equations. Depending on the system, all solutions or only a subset of all the solutions could yield a strange attractor.

Dimension

As mentioned earlier, strange attractors have non-integer dimension by definition. The dimension of a strange attractor is also limited by the dimension of the state space it inhabits. If an eight-variable system is represented by a strange attractor in state space, the dimension of the attractor is some non-integer number less than eight. The exact dimension can be calculated using the concept of fractal dimension.

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. Although not itself a strange attractor, the Cantor Set frequently shows up in the geometry of strange attractors.

References

  1. Brin, M., & Stuck, G. (2002). Introduction to dynamical systems. Cambridge; New York: Cambridge University Press.
  2. Sprott, J. (1993). Strange Attractors: creating patterns in chaos. New York: Henry Holt & Company.