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.]]
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[[Image:Strange Attractor.jpg|thumb|380px|right|A visualization of the Poisson Saturne attractor, 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 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.   
  
 
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, 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>
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If you examined the evolution of a system characterized by a strange attractor, you would notice some interesting things. As in any system with an attractor, nearby trajectories would migrate to the 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.  
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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.
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Try out the interactive animation on the right. The animation simultaneously plots three trajectories, colored red, blue, and green, following the famous [[Lorenz Attractor]].  Notice how the trajectories start out right near each other and seem to stick together for the first few seconds.  But then they start to fall out of sync, tracing out quite different paths through state space.  This is an example of exponential divergence due to the high sensitivity to initial conditions found on strange attractors.
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Although the different trajectories diverge from each other, they do not fly off randomly into state space.  Instead their movement is confined to the specific region of the strange attractor.  If left to whiz around long enough, these trajectories would trace out a detailed approximation of the structure of the Lorenz Attractor.
  
 
==Strange Structure==
 
==Strange Structure==
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]].
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Strange attractors are generated by graphing certain [[nonlinearity|nonlinear]] equations in phase space, or in some cases by taking lower-dimensional cross sections of the full attractor. 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]]. Surprisingly, equations generating a strange attractor to not have to be particualarly complex.  In fact, they can be very simple.<ref name=sourceone> Stewart, I. (1989). ''Does God Play Dice?''. Malden, MA: Blackwell Publishing ltd.</ref>
  
 
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.
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<Font Color=white>Spacing is good to have. I need a lot of spacing here. Blah blah blah blahbla</font>
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===Dimension===
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{{#iframe:http://www.bekkoame.ne.jp/~ishmnn/java/lorenz|460|470}}
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|fractal dimension]].
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<center><Font Size=1.9>Click start to trace three trajectories from nearby starting points on the Lorenz strange attractor.  Clicking he stop button will pause the trajectories, and the reset button will clear the traces.</font></center>
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===Dimension of Strange Attractors===
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As mentioned earlier, strange attractors have non-integer dimension.  Strange Attractors also have upper and lower limits to their dimension. For starters, in order to have an infinitely long trajectory that never crosses itself you need more than two dimensions. Imagine drawing a continuous curve on a single flat sheet of paper.  Eventually you would have to draw over an old part of the curve. Therefore the dimension of a strange attractor must be greater than two. Note that sometimes strange attractors are visualized using two dimensional cross-sections of their full structure, as is the case with the [[Henon Attractor| Hénon Map]] of the Hénon Attractor.
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The upper limit to the dimension of a strange attractor is just the dimension of the state space it inhabits.   
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So, if a five-variable system is represented by a strange attractor in its five-dimensional state space, the dimension of the attractor must be some non-integer number between two and five. For a description of how to calculate non-integer dimensions, check out the page on [[Fractal Dimension|fractal dimension]].
  
 
==History==
 
==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>
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The study of strange attractors began with the work of E.N. Lorenz and his 1963 paper "Deterministic non-periodic flow".<ref name=sourcetwo> 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=sourcethree> Sprott, J. (1993). ''Strange Attractors: creating patterns in chaos''. New York: Henry Holt & Company.</ref>
 
 
  
 
==Examples of Strange Attractors==
 
==Examples of Strange Attractors==

Revision as of 13:29, 7 June 2012

A visualization of the Poisson Saturne attractor, 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.

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 in any system with an attractor, nearby trajectories would migrate to the 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.

Try out the interactive animation on the right. The animation simultaneously plots three trajectories, colored red, blue, and green, following the famous Lorenz Attractor. Notice how the trajectories start out right near each other and seem to stick together for the first few seconds. But then they start to fall out of sync, tracing out quite different paths through state space. This is an example of exponential divergence due to the high sensitivity to initial conditions found on strange attractors.

Although the different trajectories diverge from each other, they do not fly off randomly into state space. Instead their movement is confined to the specific region of the strange attractor. If left to whiz around long enough, these trajectories would trace out a detailed approximation of the structure of the Lorenz Attractor.

Strange Structure

Strange attractors are generated by graphing certain nonlinear equations in phase space, or in some cases by taking lower-dimensional cross sections of the full attractor. 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. Surprisingly, equations generating a strange attractor to not have to be particualarly complex. In fact, they can be very simple.[1]

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.

Spacing is good to have. I need a lot of spacing here. Blah blah blah blahbla


Click start to trace three trajectories from nearby starting points on the Lorenz strange attractor. Clicking he stop button will pause the trajectories, and the reset button will clear the traces.


Dimension of Strange Attractors

As mentioned earlier, strange attractors have non-integer dimension. Strange Attractors also have upper and lower limits to their dimension. For starters, in order to have an infinitely long trajectory that never crosses itself you need more than two dimensions. Imagine drawing a continuous curve on a single flat sheet of paper. Eventually you would have to draw over an old part of the curve. Therefore the dimension of a strange attractor must be greater than two. Note that sometimes strange attractors are visualized using two dimensional cross-sections of their full structure, as is the case with the Hénon Map of the Hénon Attractor.

The upper limit to the dimension of a strange attractor is just the dimension of the state space it inhabits.

So, if a five-variable system is represented by a strange attractor in its five-dimensional state space, the dimension of the attractor must be some non-integer number between two and five. For a description of how to calculate non-integer dimensions, check out the page on fractal dimension.

History

The study of strange attractors began with the work of E.N. Lorenz and his 1963 paper "Deterministic non-periodic flow".[2] 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.[3]

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. Stewart, I. (1989). Does God Play Dice?. Malden, MA: Blackwell Publishing ltd.
  2. Brin, M., & Stuck, G. (2002). Introduction to dynamical systems. Cambridge; New York: Cambridge University Press.
  3. Sprott, J. (1993). Strange Attractors: creating patterns in chaos. New York: Henry Holt & Company.