Zaslavskii map

Dynamical system that exhibits chaotic behavior

{\displaystyle \epsilon =5,\nu =0.2,r=2.} — **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 (1–2): 189–208. Bibcode:1983PhyD....9..189G. doi:10.1016/0167-2789(83)90298-1. (LINK)

Chaos theory

Concepts

Core	Attractor Bifurcation Fractal Limit set Lyapunov exponent Orbit Periodic point Phase space
Anosov diffeomorphism Arnold tongue axiom A dynamical system Bifurcation diagram Box-counting dimension Correlation dimension Conservative system Ergodicity False nearest neighbors Hausdorff dimension Invariant measure Lyapunov stability Measure-preserving dynamical system Mixing Poincaré section Recurrence plot SRB measure Stable manifold Topological conjugacy
Theorems	Ergodic theorem Liouville's theorem Krylov–Bogolyubov theorem Poincaré–Bendixson theorem Poincaré recurrence theorem Stable manifold theorem Takens's theorem

Theoretical
branches

Chaotic
maps (list)

Discrete	Arnold's cat map Baker's map Complex quadratic map Coupled map lattice Duffing map Dyadic transformation Dynamical billiards outer Exponential map Gauss map Gingerbreadman map Hénon map Horseshoe map Ikeda map Interval exchange map Irrational rotation Kaplan–Yorke map Langton's ant Logistic map Standard map Tent map Tinkerbell map Zaslavskii map
Continuous	Double scroll attractor Duffing equation Lorenz system Lotka–Volterra equations Mackey–Glass equations Rabinovich–Fabrikant equations Rössler attractor Three-body problem Van der Pol oscillator