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,[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 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
- [math]\displaystyle{ K \approx S^2 = (\Delta \omega_r/\Delta_d)^2 \gt 1 . }[/math]
Here [math]\displaystyle{ K }[/math] is the perturbation parameter, while [math]\displaystyle{ S = \Delta \omega_r/\Delta_d }[/math] is the resonance-overlap parameter, given by the ratio of the unperturbed resonance width in frequency [math]\displaystyle{ \Delta \omega_r }[/math] (often computed in the pendulum approximation and proportional to the square-root of perturbation), and the frequency difference [math]\displaystyle{ \Delta_d }[/math] 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
References
External links
- website dedicated to Boris Chirikov
- Special Volume dedicated to 70th of Boris Chirikov: Physica D 131:1-4 vii (1999) and arXiv
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