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 (x_n, y_n ) in the plane and maps it to a new point given by

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

${\displaystyle y_{n+1}=\alpha y_{n}+\cos(4\pi x_{n})}$

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

Calculation method[]

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:

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

${\displaystyle x_{n+1}=a_{n}/b}$

${\displaystyle y_{n+1}=\alpha y_{n}+\cos(4\pi x_{n})}$

where the ${\displaystyle a_{n}}$ and ${\displaystyle b}$ are computational integers. It is also best to choose ${\displaystyle b}$ to be a large prime number in order to get many different values of ${\displaystyle x_{n}}$.

Another way to avoid having the modulo operator yield zero after a short number of iterations is

${\displaystyle x_{n+1}=2x_{n}\ ({\textrm {mod}}~0.99995)}$

${\displaystyle y_{n+1}=\alpha y_{n}+\cos(4\pi x_{n})}$

which will still eventually return zero, albeit after many more iterations.

References[]

• 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.

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