Chirikov criterion

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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 = Δωr / Δd is the resonance-overlap parameter, given by the ratio of the unperturbed resonance width in frequency Δωr (often computed in the pendulum approximation and proportional to the square-root of perturbation), and the frequency difference Δd between two unperturbed resonances. Since its introduction, the Chirikov criterion has become an important analytical tool for the determination of the chaos border.

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

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