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Kaplan-Yorke map

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Title: Kaplan-Yorke map  
Author: World Heritage Encyclopedia
Language: English
Subject: Dynamical system, List of chaotic maps
Publisher: World Heritage Encyclopedia

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_{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

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)\,
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.


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