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,
B. V. Chirikov, "Resonance processes in magnetic traps", At. 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.

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

See also

Chirikov criterion at Scholarpedia

Chirikov standard map
and standard map
* Boris Chirikov an
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)

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) an
arXiv

Chaos theory
Chaotic maps

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