Zaslavskii map

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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.

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