Transcription of WHAT IS a Strange Attractor?
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WHAT IS a Strange Attractor?
A Strange Attractor? David Ruelle764 NOTICESOFTHEAMSVOLUME53, NUMBER7 Your computer will readily implement the map fsending the point (u,v) R2to the point(v+1 au2,bu) R2, where a= b= your computer to plot the points xn=fn(0,0),and you will find that they accumulate, for n ,on a convoluted fractal set Aknown as the H nonattractor. This set is prototypical of what one wantsto call a Strange attractor . Such objects often arisewhen a diffeomorphism fstretches and folds anopen set Uand maps the closure f Uinside U(thisis a typical situation, not a necessary or a sufficientcondition). The Strange attractor Ais visualizedwhen a computer plots the points xn=fnx0withalmost any initial value x0in U. The figures showa two-dimensional example corresponding to theH non attractor , and a three-dimensional examplecorresponding to Smale s solenoid. It turns out thata small change in the values of aand bcan destroythe H non attractor , but a small change of fdoesnot destroy the give now a definition of uniformly hyperbolicattractors, or Axiom A attractors.
a Strange Attractor? David Ruelle 764 NOTICES OF THE AMS VOLUME 53, NUMBER 7 Your computer will readily implement the map f sending the point (u,v)∈ R2 to the point(v +1−au2,bu)∈ R2, where a =1.4and b =0.3. Ask your computer to plot the points xn =fn(0,0), and you will find that they accumulate, for n →∞, on a convoluted fractal set A known as the Hénon ...
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