Kaplan–Yorke map
# 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 $(x_n,$ $y_n$ ) 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.

## 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:

    $a_{n+1}=2a_{n}\ ({\textrm {mod}}~b)$
    $x_{n+1}=a_{n}/b$
    $y_{n+1}=\alpha y_{n}+\cos(4\pi x_{n})$

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 $x_{n}$.

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

    $x_{n+1}=2x_{n}\ ({\textrm {mod}}~0.99995)$
    $y_{n+1}=\alpha y_{n}+\cos(4\pi x_{n})$

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

