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.

References

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: 189–208. Bibcode:1983PhyD....9..189G. doi:10.1016/0167-2789(83)90298-1. (LINK)

Chaos theory

Chaos theory	Anosov diffeomorphism Bifurcation theory Butterfly effect Chaos theory in organizational development Complexity Control of chaos Dynamical system Edge of chaos Fractal Predictability Quantum chaos Santa Fe Institute Synchronization of chaos Unintended consequences

Chaotic maps (list)	Arnold tongue Arnold's cat map Baker's map Complex quadratic map Complex squaring map Coupled map lattice Double pendulum Double scroll attractor Duffing equation Duffing map Dyadic transformation Dynamical billiards outer Exponential map Gauss map Gingerbreadman map Hénon map Horseshoe map Ikeda map Interval exchange map Kaplan–Yorke map Logistic map Lorenz system Multiscroll attractor Rabinovich–Fabrikant equations Rössler attractor Standard map Swinging Atwood's machine Tent map Tinkerbell map Van der Pol oscillator Zaslavskii map

Chaos systems	Bouncing ball dynamics Chua's circuit Economic bubble Tilt-A-Whirl

Chaos theorists	Michael Berry Leon O. Chua Mitchell Feigenbaum Martin Gutzwiller Brosl Hasslacher Michel Hénon Edward Norton Lorenz Aleksandr Lyapunov Benoît Mandelbrot Henri Poincaré Otto Rössler David Ruelle Oleksandr Mykolaiovych Sharkovsky Floris Takens James A. Yorke

This article is issued from Wikipedia - version of the 6/30/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.