Zaslavskii map

Dynamical system that exhibits chaotic behavior

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Zaslavskii map with parameters:

\epsilon =5,\nu =0.2,r=2.

{\displaystyle \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 ( $x_{n},y_{n}$ {\displaystyle 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})],円({\textrm {mod}},1円)

{\displaystyle x_{n+1}=[x_{n}+\nu (1+\mu y_{n})+\epsilon \nu \mu \cos(2\pi x_{n})],円({\textrm {mod}},1円)}

y_{n+1}=e^{-r}(y_{n}+\epsilon \cos(2\pi x_{n})),円

{\displaystyle y_{n+1}=e^{-r}(y_{n}+\epsilon \cos(2\pi x_{n})),円}

and

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

{\displaystyle \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.

References

[edit ]

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

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