Zaslavsky map

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George Zaslavsky (2007), Scholarpedia, 2(5):2662. doi:10.4249/scholarpedia.2662 revision #91950 [link to/cite this article]
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Curator: George Zaslavsky

Zaslavsky map (also known as the "Zaslavskii Map", the dissipative kicked rotor, or the dissipative standard map) occurs in the literature in two very different situations: as the dissipative generalization of the standard map (Zaslavsky 1978, Zaslavsky and Rachko 1979) or as the web map (Zaslavsky et al. 1986, Zaslavsky et al. 1991). Here, only the first application is described.


The Map

Consider a system in action-angle \((I,x)\) variables that is assumed to have a stable limit cycle at \( I=1 \ .\) Now perturb this system, kicking it periodically; the equations of motion are \[ \dot{I} = -\Gamma (I-1)+\varepsilon \sin x \sum_{n=-\infty}^\infty \delta (t-n); \ \ \dot{x}=\omega(I), \] where \( \varepsilon \) is a perturbation parameter, \( \Gamma >0 \ ,\) the period of the kicks is 1, and \(\omega\) is the nonlinear frequency. A simplified version of the frequency is \[\omega(I)=\Omega+\alpha y; \ \ y=I-1 \ .\] These equations can be integrated between successive kicks, to give the time-one map, the Zaslavsky Map: \[y_{n+1}=e^{-\Gamma}(y_n+\varepsilon \sin x_n) \ ;\] \[\tag{1} x_{n+1}=x_n+ \Omega + \alpha \mu y_{n+1}, \ \ \mod 2 \pi; \ \ \mu =(e^{\Gamma}-1)/\Gamma\ ,\]

This system is also called the dissipative kicked rotor, or dissipative standard map.

Qualitatively, the system (1) can be described as rotation along the limit cycle defined by the unperturbed equation \( \dot{I} = -\Gamma (I-1)\) that is periodically perturbed by the kicks that change the actual radius of rotation. After each kick the trajectory relaxes back toward the stable limit cycle until the next kick knocks it off again.

Chaotic Attractors

Figure 1: Chaotic attractor generated by the Zaslavsky Map on \((p,x)\) plane.

The solutions of the map (1) depend on the competition between the dissipation factor \( \Gamma \) and the perturbation parameter \( \varepsilon \alpha \ .\) A chaotic attractor appears when the combination \( K =\epsilon \alpha (1-\exp(-\Gamma))/\Gamma \stackrel {>}{\sim} 1 \) is fairly large (Zaslavsky 1978, Zaslavsky and Rachko 1979). These attractors have a structure like that shown in Figure 1. Each "quasi-line" shown in the figure is thick and has a fractal type filaments typical for all fractal attractors. The number of "quasi-lines" depends on \( K \ .\) Near the threshold at which the attractor disappears, the fractal structure of the attractor is not as evident (see e.g. Figure 2).

Upon defining new variables \((p,x)\ ,\) by scaling \(p=(\alpha \mu)y\ ,\) the map (1) takes a form that resembles the standard map: \[p_{n+1}=Ap_n+K \sin x_n \ ;\] \[x_{n+1}=x_n+ \Omega + p_{n+1}, \ \ \mod 2 \pi. \] However, because \(A=\exp(-\Gamma)<1\ ,\) whenever \(\Gamma \ne 0\ ,\) the map is not area preserving: indeed the Jacobian \(|\partial(p_{n+1};x_{n+1})/\partial(p_{n};x_{n})|=A<1\ .\)

Figure 2: Dying chaotic attractor for the Zaslavsky Map.

One-Dimensional Map

A \( \Gamma \) becomes large some properties of (1) can be studied by reducing the two-dimensional map to an approximate one-dimensional map, the Arnold circle map that has been extensively studied \[x_{n+1}=x_n+ \Omega + K \sin x_{n}, \ \ \mod 2 \pi\ .\]

Studies of the Zaslavsky Map

The fractal dimension of the chaotic attractor generated by (1) was studied by Russel, Hanson, and Ott in 1980 and by Grassberg and Procaccia in 1983. A rigorous theory of systems like (1) that have a strong contraction to a nearly one-dimensional system has been given by Wang and Young in 2002. Their article also gives a classification of different types of solutions that can occur in (1). An extension of the classical the Zaslavsky Map to the Quantum realm was studied by Haake in 2000.

The Zaslavsky Map can also appear as an important model in dynamical systems when a saddle-node bifurcation is subjected to small periodic perturbations (see Arnold et al. 1994, Afraimovich and Hsu 2003).


  • Arnold V.I., Afraimovich V.S., Il'yashenko Yu.S., and Shilnikov V.P. (1994) "Encyclopedia of Mathematical Sciences, Dynamical Systems V". Springer, Berlin
  • Afraimovich V. and Hsu Sze-Bi (2003) "Lectures in Chaotic Dynamical Systems". American Mathematical Society. Interpress.
  • Grassberger P. and Procaccia I. (1983) Measuring the strangeness of strange attractors. Physica D 9:189-208
  • Haake F. (2000) Quantum signatures of Chaos. Springer, Berlin
  • Russell D.A., Hanson J.D., and Ott E. (1980) Dimension of strange attractors. PRL 45:1175-1178
  • Wang Q. and Young L.-S. (2002) From invariant curves to strange attractors. Commun. in Math. Phys. 225:275-304
  • Zaslavsky G.M. (1978) The simplest case of a strange attractor. Phys. Lett. A, 69:145-147
  • Zaslavsky G.M. and Rachko Kh.-R.Ya (1979) Singularities of transition to a turbulent motion. Sov. Phys. JETP 49:1039-1044
  • Zaslavsky G.M., Zakharov M.Yu., Sagdeev R.Z., Usikov D.A, and Chernikov A.A. (1986) Stochastic web and diffusion of particles in a magnetic field. Sov. Phys. JETP 64:294:303
  • Zaslavsky G.M., Sagdeev R.Z., Usikov D.A, and Chernikov A.A. (1991) Weak chaos and Quasiregular Patterns. Cambridge Univ. Press, Cambridge

Internal references

See Also

Chaos, Dynamical Systems, Iterated Maps, Quasiperiodic Oscillations, Fractal, Circle Map, Standard Map

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