Difference between revisions of "Strange Attractors"
(10 intermediate revisions by the same user not shown)  
Line 1:  Line 1:  
−  [[Image:Strange Attractor.jpgthumb  +  <div style="float: left; width: 24%"> 
+  
+  
+  ==<font color=white>hello</font>==  
+  </div><div style="float: left; width: 36%">  
+  [[Image:Strange Attractor.jpgthumb335pxA visualization of the Poisson Saturne strange attractor.]]  
+  </div><div style="float: right; width: 39%">  
+  {{HelperPage1=Lorenz Attractor2=Henon Attractor3=Field:Fractals4=Field:Dynamic Systems5=Chaos}}  
+  <font color=white>hello.  
+  
+  
+  More  
+  
+  f  
+  
+  
+  i  
+  
+  
+  
+  l  
+  
+  
+  l  
+  
+  e  
+  
+  
+  r  
+  </font>  
+  </div>  
+  
==Basic Description==  ==Basic Description==  
−  A '''strange attractor''', or '''chaotic attractor''', is an infinitepoint [[Field:Dynamic Systems#Jump3attractor]] with [[Fractal Dimensionnoninteger dimension]]. Although they consist of an infinite number of points, strange attractors do not fill [[Field:Dynamic Systems#Jump2state space]]. Instead, they are contained within a bounded region and are  +  A '''strange attractor''', or '''chaotic attractor''', is an infinitepoint [[Field:Dynamic Systems#Jump3attractor]] with [[Fractal Dimensionnoninteger dimension]]. Although they consist of an infinite number of points, strange attractors do not fill [[Field:Dynamic Systems#Jump2state space]]. Instead, they are contained within a bounded region and are highly structured. In fact, strange attractors are a type of [[Field:Fractalsfractal]], exhibiting selfsimilarity. 
+  
+  In [[Field:Dynamic Systemsdynamical systems theory]], the dynamics of [[Chaoschaotic]] systems are represented by strange attractors.  
+  
+  ==Strange Trajectories==  
+  <div style="float: left; width: 57%">  
+  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 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 Growthexponentially]]. 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, trajectories following a strange attractor can be infinitely long, but never repeat themselves. 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. Nonperiodic systems have no predictable cycles.">nonperiodic</balloon>.  
+  
+  Try out the interactive animation on the right. The animation simultaneously plots three trajectories, colored red, blue, and green, following the famous [[Lorenz AttractorLorenz strange 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 with 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 region around the strange attractor. If left to whiz around long enough, these trajectories would trace out an approximation of the structure of the Lorenz Attractor.  
+  
+  ==Strange Structure==  
+  Strange attractors are generated by certain [[nonlinearitynonlinear]] equations, and can be visualized by plotting the longterm trajectories described by these equations in phase space. In some cases strange attractors are visualized using lowerdimensional crosssections of their full trajectories, as is the case with the Hénon Map of the Hénon 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 AttractorHénon 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>  
+  
+  As mentioned earlier, strange attractors have noninteger dimension. For a description of how to calculate noninteger dimensions, check out the page on [[Fractal Dimensionfractal dimension]].  
+  </div><div style="float: left; width: 2%">  
+  <Font Color=white>Spacing is good to have. I need a lot of spacing here. Blah blah blah blahbla</font>  
+  </div><div style="float: left; width: 41%">  
+  
−  +  {{#iframe:http://www.bekkoame.ne.jp/~ishmnn/java/lorenz460470}}  
−  =  +  <center><Font Size=1.9>Click ''Start'' to trace three trajectories from nearby starting points on the Lorenz strange attractor. ''Stop'' will pause the trajectories, and ''Reset'' will clear the traces. Applet created by Ishihama Yoshiaki. 
−  
−  
−  
+  </font></center>  
+  
+  </div>  
==History==  ==History==  
−  The study of strange attractors began with the work of E.N. Lorenz and his 1963 paper "Deterministic nonperiodic flow".<ref name=  +  The study of strange attractors began with the work of E.N. Lorenz and his 1963 paper "Deterministic nonperiodic 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 the dynamics of 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==  
−  Examples of strange attractors include the [[Henon Attractor Hénon  +  Examples of strange attractors include the [[Henon Attractor 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==  ==References==  
<references />  <references /> 
Latest revision as of 09:22, 11 July 2012
Contents
hello
This is a Helper Page for:


Lorenz Attractor 
Henon Attractor 
Field:Fractals 
Field:Dynamic Systems 
Chaos 
hello.
More
f
i
l
l
e
r
Basic Description
A strange attractor, or chaotic attractor, is an infinitepoint attractor with noninteger 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 selfsimilarity.
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 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.
Furthermore, trajectories following a strange attractor can be infinitely long, but never repeat themselves. 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 nonperiodic.
Try out the interactive animation on the right. The animation simultaneously plots three trajectories, colored red, blue, and green, following the famous Lorenz strange 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 with 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 region around the strange attractor. If left to whiz around long enough, these trajectories would trace out an approximation of the structure of the Lorenz Attractor.
Strange Structure
Strange attractors are generated by certain nonlinear equations, and can be visualized by plotting the longterm trajectories described by these equations in phase space. In some cases strange attractors are visualized using lowerdimensional crosssections of their full trajectories, as is the case with the Hénon Map of the Hénon 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]}
As mentioned earlier, strange attractors have noninteger dimension. For a description of how to calculate noninteger dimensions, check out the page on fractal dimension.
Spacing is good to have. I need a lot of spacing here. Blah blah blah blahbla
History
The study of strange attractors began with the work of E.N. Lorenz and his 1963 paper "Deterministic nonperiodic 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 the dynamics of 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.