Zaslavskii map

Short description: Dynamical system that exhibits chaotic behavior

Zaslavskii map with parameters: [math]\displaystyle{ \epsilon=5, \nu=0.2, r=2. }[/math]

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 ([math]\displaystyle{ x_n,y_n }[/math]) in the plane and maps it to a new point:

[math]\displaystyle{ x_{n+1}=[x_n+\nu(1+\mu y_n)+\epsilon\nu\mu\cos(2\pi x_n)]\, (\textrm{mod}\,1) }[/math]

[math]\displaystyle{ y_{n+1}=e^{-r}(y_n+\epsilon\cos(2\pi x_n))\, }[/math]

and

[math]\displaystyle{ \mu = \frac{1-e^{-r}}{r} }[/math]

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.

See also

• List of chaotic maps

References

• G.M. Zaslavskii (1978). "The Simplest case of a strange attractor". Phys. Lett. A 69 (3): 145–147. doi:10.1016/0375-9601(78)90195-0. Bibcode: 1978PhLA...69..145Z.  (LINK)
• D.A. Russel; J.D. Hanson; E. Ott (1980). "Dimension of strange attractors". Phys. Rev. 45 (14): 1175. doi:10.1103/PhysRevLett.45.1175. Bibcode: 1980PhRvL..45.1175R.  (LINK)
• P. Grassberger and I. Procaccia (1983). "Measuring the strangeness of strange attractors". Physica 9D (1–2): 189–208. doi:10.1016/0167-2789(83)90298-1. Bibcode: 1983PhyD....9..189G.  (LINK)