Zaslavsky map
| George Zaslavsky (2007), Scholarpedia, 2(5):2662. | doi:10.4249/scholarpedia.2662 | revision #91950 [link to/cite this article] |
Zaslavsky map (ZM, also known as "Zaslavskii Map") occurs in the literature for two very different objects: dissipative generalization of the standard map (Zaslavsky 1978, Zaslavsky and Rachko 1979) and web map (Zaslavsky et al. 1986, Zaslavsky et al. 1991). Here, only the first one is described.
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The Map
The unperturbed system is considered in action-angle \((I,x)\) variables and it is assumed to have a stable limit cycle at \( I=1 \). When perturbed by periodic kicks, 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 for kicks equals 1 and \(\omega\) is the nonlinear frequency. A simplified version for the frequency is \[\omega(I)=\Omega+\alpha y; \ \ y=I-1 \]. These equations, being written as a map between successive kicks, give the Zaslavsky Map: \[y_{n+1}=e^{-\Gamma}(y_n+\varepsilon \sin x_n) \]; \[x_{n+1}=x_n+ \Omega + \alpha \mu y_{n+1}, \ \ \mod 2 \pi; \ \ \mu =(e^{\Gamma}-1)/\Gamma\], which is also called the dissipative kicked rotor model, or dissipative standard map.
The main property of the first system can be described, in a qualitative way, as rotation along the limit cycle defined by the unperturbed equation \( \dot{I} = -\Gamma (I-1)\) and periodic perturbation from the kicks that change the actual radius of rotation. After each kick the trajectory moves back toward the stable limit cycle until the next kick takes it off again.
Chaotic Attractors
The solutions of the Zaslavsky Map depend on the competition between dissipation factor \( \Gamma \) and perturbation parameter \( \varepsilon \alpha \). Chaotic attractor appears for fairly large \(K =\epsilon \alpha (1-\exp(-\Gamma))/\Gamma \stackrel {>}{\sim} 1\) (Zaslavsky 1978, Zaslavsky and Rachko 1979). The attractor has a structure shown in Figure <ref>F1</ref>. Each "quasi-line" is thick and has a fractal type filaments typical for all chaotic attractors. The number of "quasi-lines" depends on \( K \). Near a threshold of the attractor disappearance, the structure of the attractor is poor ("dying chaotic attractor" in Figure <ref>F2</ref>).
On the plane \((p,x)\), \(p=(\alpha \mu)y\), Zaslavsky Map has a fairly simple form that resembles the standard map: \[p_{n+1}=Ap_n+K \sin x_n \]; \[x_{n+1}=x_n+ \Omega + p_{n+1}, \ \ \mod 2 \pi. \] Because \(A=\exp(-\Gamma)<1\) if \(\Gamma \ne 0\), the map is not area preserving and the Jacobian \(|\partial(p_{n+1};x_{n+1})/\partial(p_{n};x_{n})|=A<1\).
One-Dimensional Map
For fairly large \( \Gamma \) some properties of the Zaslavsky Map can be studied by reducing the 2-dimensional map to an approximate one-dimensional map, the circle map that has been extensively studied \[x_{n+1}=x_n+ \Omega + K \sin x_{n}, \ \ \mod 2 \pi\].
Fractal dimension of chaotic attractor generated by the Zaslavsky Map was studied in Russel, Hanson, and
Ott 1980 and Grassberg and Procaccia 1983. Rigorous theory of the systems
of the Zaslavsky Map - type, that are close to one-dimensional maps, is in Wang and
Young 2002. In this article there is also a classification of different
types of solutions for the system of the Zaslavsky Map - type. Quantum consideration of
the Zaslavsky Map is in Haake 2000.
Zaslavsky Map can appear as an important model in dynamical systems with the saddle-node bifurcation subjected to small periodic perturbations ( see Arnold et al. 1994, Afraimovich and Hsu 2003).
References
- 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
See Also
Chaos, Dynamical Systems, Iterated Maps, Quasiperiodic Oscillations
