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# Kaplan-Yorke map

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 Title: Kaplan-Yorke map Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Kaplan-Yorke map

The Kaplan–Yorke map is a discrete-time dynamical system. It is an example of a dynamical system that exhibits chaotic behavior. The Kaplan–Yorke map takes a point (xn, yn ) in the plane and maps it to a new point given by

$x_\left\{n+1\right\}=2x_n\ \left(\textrm\left\{mod\right\}~1\right)\,$
$y_\left\{n+1\right\}=\alpha y_n+\cos\left(4\pi x_n\right)\,$

where mod is the modulo operator with real arguments. The map depends on only the one constant α.

## Calculation method

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:

$a_\left\{n+1\right\}=2a_n\ \left(\textrm\left\{mod\right\}~b\right)\,$
$x_\left\{n+1\right\}=a_n/b\,$
$y_\left\{n+1\right\}=\alpha y_n+\cos\left(4\pi x_n\right)\,$

where the $a_n$ and $b$ are computational integers. It is also best to choose $b$ to be a large prime number in order to get many different values of $x_n$.

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