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 (Atom. Energ. 6: 630 (1959)), 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. As in an old oriental tale, Boris Chirikov opened such a bottle, and freed the genie of chaos, which spread the world over.

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 = Δω_r / Δ_d is the resonance-overlap parameter, given by the ratio of the unperturbed resonance width in frequency Δω_r (often computed in the pendulum approximation and proportional to the square-root of perturbation), and the frequency difference Δ_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, "Resonance processes in magnetic traps", At. Energ. 6: 630 (1959) (in Russian [1])(Engl. Transl., J. Nucl. Energy Part C: Plasma Phys. 1: 253 (1960)[2])
• 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