Chirikov criterion
Encyclopedia
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.
between two nonlinear resonance
s in a chaotic and unpredictable manner,
in the parameter range
K \approx S^2 = (\Delta \omega_r/\Delta_d)^2 > 1 .
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.
was established by the Russian physicist Boris Chirikov
Boris Chirikov
Boris Valerianovich Chirikov was an outstanding Soviet and Russian physicist....
.
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
Kurchatov Institute
The Kurchatov Institute is Russia's leading research and development institution in the field of nuclear energy. In the Soviet Union it was known as I. V. Kurchatov Institute of Atomic Energy , abbreviated KIAE . It is named after Igor Kurchatov....
.
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 movebetween two nonlinear resonance
Nonlinear resonance
In physics, nonlinear resonance is the occurrence of resonance in a nonlinear system. In nonlinear resonance the system behaviour – resonance frequencies and modes – depends on the amplitude of the oscillations, while for linear systems this is independent of amplitude.-Description:Generically two...
s in a chaotic and unpredictable manner,
in the parameter range
K \approx S^2 = (\Delta \omega_r/\Delta_d)^2 > 1 .
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 ChirikovBoris ChirikovBoris Valerianovich Chirikov was an outstanding Soviet and Russian physicist....
and Boris Chirikov at Scholarpedia
External links
- website dedicated to Boris Chirikov
- Special Volume dedicated to 70th of Boris Chirikov: Physica D 131:1-4 vii (1999) and arXiv