# Chirikov criterion

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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

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

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

## 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