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| [[Image:Kaplan-Yorke map.png|thumb|A plot of 100,000 iterations of the Kaplan-Yorke map with α=0.2. The initial value (x<sub>0</sub>,y<sub>0</sub>) was (128873/350377,0.667751).]]
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| The '''Kaplan–Yorke map''' is a [[discrete-time]] [[dynamical system]]. It is an example of a dynamical system that exhibits [[chaos theory|chaotic behavior]]. The Kaplan–Yorke [[function (mathematics)|map]] takes a point (''x<sub>n</sub>, y<sub>n</sub> '') in the [[plane (mathematics)|plane]] and [[function (mathematics)|maps]] it to a new point given by
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| :<math>x_{n+1}=2x_n\ (\textrm{mod}~1)\,</math>
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| :<math>y_{n+1}=\alpha y_n+\cos(4\pi x_n)\,</math>
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| where ''mod'' is the [[modulo operation|modulo operator]] with real arguments. The map depends on only the one [[Constant (mathematics)|constant]] α.
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| ==Calculation method==
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| Due to roundoff error, successive applications of the modulo operator will yield zero after some ten or twenty iterations when implemented as a floating point operation on a computer. It is better to implement the following equivalent algorithm:
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| :<math>a_{n+1}=2a_n\ (\textrm{mod}~b)\,</math> | |
| :<math>x_{n+1}=a_n/b\,</math> | |
| :<math>y_{n+1}=\alpha y_n+\cos(4\pi x_n)\,</math>
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| where the <math>a_n</math> and <math>b</math> are computational integers. It is also best to choose <math>b</math> to be a large [[prime number]] in order to get many different values of <math>x_n</math>.
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| ==References==
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| * {{cite book | author= J.L. Kaplan and J.A. Yorke | title= Functional Differential Equations and Approximations of Fixed Points (Lecture notes in Mathematics 730) |editor=H.O. Peitgen and H.O. Walther | publisher=Springer-Verlag | year=1979 | isbn = 0-387-09518-7}}
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| * {{cite journal | author=P. Grassberger and I. Procaccia | title=Measuring the strangeness of strange attractors | journal=Physica | year=1983 | volume=9D | issue=1-2| pages=189–208 | bibcode=1983PhyD....9..189G | doi=10.1016/0167-2789(83)90298-1 }}
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| {{Chaos theory}}
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| {{DEFAULTSORT:Kaplan Yorke Map}}
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| [[Category:Chaotic maps]]
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| {{Mathapplied-stub}}
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| {{physics-stub}}
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Nothing to say about me I think.
I enjoy of finally being a member of this site.
I just hope Im useful in some way .
Feel free to surf to my homepage :: Fifa 15 Coin Generator