Kaplan-Yorke map

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A plot of 100,000 iterations of the Kaplan-Yorke map with α=0.2. The initial value (x₀,y₀) was (128873/350377,0.667751).

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

$x_{n+1}=2x_n\ (\textrm{mod}~1)\,$

$y_{n+1}=\alpha y_n+\cos(4\pi x_n)\,$

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

[edit] 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:

$a_{n+1}=2a_n\ (\textrm{mod}~b)\,$

$x_{n+1}=a/b\,$

$y_{n+1}=\alpha y_n+\cos(4\pi x_n)\,$

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 $x n$ .

[edit] References

J.L. Kaplan and J.A. Yorke (1979). in H.O. Peitgen and H.O. Walther: Functional Differential Equations and Approximations of Fixed Points (Lecture notes in Mathematics 730). Springer-Verlag. ISBN 0-387-09518-7.
P. Grassberger and I. Procaccia (1983). "(LINK) Measuring the strangeness of strange attractors". Physica 9D: 189-208.