Dynamical system that exhibits chaotic behavior
Zaslavskii map with parameters: ![{\displaystyle \epsilon =5,\nu =0.2,r=2.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab967b391e190833a650601e3a50d3d524f68dc5)
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 (
) in the plane and maps it to a new point:
![{\displaystyle x_{n+1}=[x_{n}+\nu (1+\mu y_{n})+\epsilon \nu \mu \cos(2\pi x_{n})]\,({\textrm {mod))\,1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/adfd5ccf515d6ae89e6340eb9ec4775d8ef23e40)
![{\displaystyle y_{n+1}=e^{-r}(y_{n}+\epsilon \cos(2\pi x_{n}))\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7bac19db5f2349ac9224b4cecbe514afd05a1a01)
and
![{\displaystyle \mu ={\frac {1-e^{-r)){r))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95ac28e99d9e4e84a80540c484756e485f1bf351)
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.