Kaplan-Yorke map

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

:$x\_\{n+1\}=2x\_n\; (\; extrm\{mod\}~1),$:$y\_\{n+1\}=alpha\; y\_n+cos(4pi\; x\_n),$

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:

:$a\_\{n+1\}=2a\_n\; (\; extrm\{mod\}~b),$:$x\_\{n+1\}=a/b,$:$y\_\{n+1\}=alpha\; y\_n+cos(4pi\; x\_n),$

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 $x\_n$.

References

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