Chirikov criterion

The Chirikov criterion or Chirikov resonance-overlap criterion was established by the Russian physicist Boris Chirikov. Back in 1959, he published a seminal article, 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 explain puzzling experimental results on plasma confinement in magnetic bottles obtained by Rodionov at the Kurchatov Institute.

Description

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

$K\approx S^{2}=(\Delta \omega _{r}/\Delta _{d})^{2}>1.$

Here $K$ is the perturbation parameter, while $S=\Delta \omega _{r}/\Delta _{d}$ is the resonance-overlap parameter, given by the ratio of the unperturbed resonance width in frequency $\Delta \omega _{r}$ (often computed in the pendulum approximation 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)
• A.J.Lichtenberg and M.A.Lieberman (1992). Regular and Chaotic Dynamics. Springer, Berlin. ISBN 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