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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 resonance]]s in a chaotic and unpredictable manner, | |||
in the parameter range | |||
:<math mylabel1> | |||
K \approx S^2 = (\Delta \omega_r/\Delta_d)^2 > 1 . | |||
</math> | |||
Here <span style="vertical-align:+4%"><math> K</math></span> is the perturbation parameter, | |||
while | |||
<span style="vertical-align:+8%"><math> S = \Delta \omega_r/\Delta_d</math></span> | |||
is the resonance-overlap parameter, given by the ratio of the | |||
unperturbed resonance width in frequency | |||
<span style="vertical-align:+8%"><math> \Delta \omega_r</math></span> | |||
(often computed in the pendulum | |||
approximation and proportional to the square-root of perturbation), | |||
and the frequency difference | |||
<span style="vertical-align:+8%"><math> \Delta_d</math></span> | |||
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== | |||
* [http://www.scholarpedia.org/article/Chirikov_criterion Chirikov criterion at Scholarpedia] | |||
* [http://www.scholarpedia.org/article/Chirikov_standard_map Chirikov standard map] and [[standard map]] | |||
* [[Boris Chirikov]] and [http://www.scholarpedia.org/article/Boris_Chirikov Boris Chirikov at Scholarpedia] | |||
==References== | |||
* B.V.Chirikov, "Resonance processes in magnetic traps", At. Energ. 6: 630 (1959) (in Russian [http://www.quantware.ups-tlse.fr/chirikov/refs/chi1959.pdf])(Engl. Transl., J. Nucl. Energy Part C: Plasma Phys. 1: 253 (1960)[http://www.quantware.ups-tlse.fr/chirikov/refs/chi1959e.pdf]) | |||
* 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) | |||
*{{cite book | author=A.J.Lichtenberg and M.A.Lieberman | title=Regular and Chaotic Dynamics | publisher=Springer, Berlin | year=1992 |isbn=978-0-387-97745-4}} [http://www.springer.com/math/analysis/book/978-0-387-97745-4 Springer link] | |||
==External links== | |||
* [http://www.quantware.ups-tlse.fr/chirikov/ website dedicated to Boris Chirikov] | |||
* Special Volume dedicated to 70th of Boris Chirikov: Physica D 131:1-4 vii (1999) and [http://arxiv.org/abs/cond-mat/9903412 arXiv] | |||
[[Category:Chaos theory]] | |||
[[Category:Chaotic maps]] | |||
Revision as of 19:01, 21 April 2013
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
Here is the perturbation parameter, while is the resonance-overlap parameter, given by the ratio of the unperturbed resonance width in frequency (often computed in the pendulum approximation and proportional to the square-root of perturbation), and the frequency difference 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)
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 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