Chirikov criterion explained

The Chirikov criterion or Chirikov resonance-overlap criterionwas established by the Russian physicist Boris Chirikov.Back in 1959, he published a seminal article,^[1] where he introduced the very first physical criterion for the onset of chaotic motion in deterministic Hamiltonian systems. He then applied such a criterion to explainpuzzling experimental results on plasma confinement in magnetic bottlesobtained by Rodionov at the Kurchatov Institute.

Description

According to this criterion a deterministic trajectory will begin to movebetween two nonlinear resonances in a chaotic and unpredictable manner, in the parameter range

$K\; \backslash approx\; S^2\; =\; (\backslash Delta\; \backslash omega\_r/\backslash Delta\_d)^2\; >\; 1\; .$Here

K is the perturbation parameter,while S = \Delta \omega _{r/\Delta} _{d} is the resonance-overlap parameter, given by the ratio of theunperturbed resonance width in frequency\Delta \omega _{r} (often computed in the pendulumapproximation and proportional to the square-root of perturbation),and the frequency difference \Delta _{d} between two unperturbed resonances. Since its introduction, the Chirikov criterion has become an important analytical tool for the determination of the chaos border.

See also

• Chirikov criterion at Scholarpedia
• Chirikov standard map and standard map
• Boris Chirikov and Boris Chirikov at Scholarpedia

References

• B.V.Chirikov, "Research concerning the theory of nonlinear resonance and stochasticity", Preprint N 267, Institute of Nuclear Physics, Novosibirsk (1969), (Engl. Trans., CERN Trans. 71-40 (1971))
• B.V.Chirikov, "A universal instability of many-dimensional oscillator systems", Phys. Rep. 52: 263 (1979)
• Book: A.J.Lichtenberg and M.A.Lieberman . Regular and Chaotic Dynamics . Springer, Berlin . 1992 . 978-0-387-97745-4. Springer link

External links

• website dedicated to Boris Chirikov
• Special Volume dedicated to 70th of Boris Chirikov: Physica D 131:1-4 vii (1999) and arXiv

Notes and References

1. http://www.quantware.ups-tlse.fr/chirikov/refs/chi1959.pdf