# Zaslavskii map

Zaslavskii map with parameters: $\epsilon=5, \nu=0.2, r=2.$

The Zaslavskii map is a discrete-time dynamical system introduced by George M. Zaslavsky. It is an example of a dynamical system that exhibits chaotic behavior. The Zaslavskii map takes a point ($x_n,y_n$) in the plane and maps it to a new point:

$x_{n+1}=[x_n+\nu(1+\mu y_n)+\epsilon\nu\mu\cos(2\pi x_n)]\, (\textrm{mod}\,1)$
$y_{n+1}=e^{-r}(y_n+\epsilon\cos(2\pi x_n))\,$

and

$\mu = \frac{1-e^{-r}}{r}$

where mod is the modulo operator with real arguments. The map depends on four constants ν, μ, ε and r. Russel (1980) gives a Hausdorff dimension of 1.39 but Grassberger (1983) questions this value based on their difficulties measuring the correlation dimension.