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 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_{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.

Calculation method[edit]

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_n/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.

References[edit]

  • J.L. Kaplan and J.A. Yorke (1979). H.O. Peitgen and H.O. Walther, ed. 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). "Measuring the strangeness of strange attractors". Physica 9D (1-2): 189–208. Bibcode:1983PhyD....9..189G. doi:10.1016/0167-2789(83)90298-1.