The Zaslavskii Map

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The Zaslavskii map arose from consideration of the problem of a dissipative kicked rotor; it exhibits chaotic behavior. It is claimed to be the simplest problem producing a strange attractor. The approach described involves the so-called action-angle variables of Hamiltonian dynamics (in which frequencies can be obtained without actually solving the equations of motion).

Contributed by: Enrique Zeleny (November 2014)
Open content licensed under CC BY-NC-SA


Snapshots


Details

Start with the equations of the perturbed system in the variables and :

,

,

where is the action for the stable limit cycle (the dimensionless parameter of the perturbation) and is the periodic pulse.

Choosing

,

,

integrating between successive pulses, and making the substitutions

,

,

,

,

the mapping can be rewritten as

,

,

with

.

References

[1] G. M. Zaslavsky, "The Simplest Case of a Strange Attractor," Physics Letters A, 69(3), 1978 pp. 145–147. doi:10.1016/0375-9601(78)90195-0.

[2] G. Zaslavsky, "Zaslavsky Map," Scholarpedia, 2(5):2662, 2007. doi:10.4249/scholarpedia.2662.

[3] G. Hanchinamani and L. Kulakarni, "Image Encryption Based on 2-D Zaslavskii Chaotic Map and Pseudo Hadmard Transform," International Journal of Hybrid Information Technology, 7(4), 2014 pp. 185–200. www.sersc.org/journals/IJHIT/vol7_no4_ 2014/16.pdf.



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