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Special Issue: On the 80th birthday of professor A. Chenciner
On Phase at a Resonance in Slow-Fast Hamiltonian Systems
Yuyang Gaoa, Anatoly Neishtadta, Alexey Okunevb a Dept. of Math. Sciences, Loughborough University,
Loughborough, LE11 3TU Leicestershire, UK
b Pennsylvania State University, State College,
16802 Pennsylvania, United States
Abstract:
We consider a slow-fast Hamiltonian system with one fast angle variable (a fast phase) whose frequency vanishes on some surface
in the space of slow variables (a resonant surface). Systems of such form appear in the study of dynamics of charged particles
in an inhomogeneous magnetic field
under the influence of high-frequency electrostatic waves. Trajectories of the system averaged over the fast phase cross the resonant surface.
The fast phase makes $\sim \frac 1\varepsilon$ turns before arrival at the resonant surface ($\varepsilon$
is a small parameter of the problem). An asymptotic formula for the value of the phase at the arrival at the resonance
was derived earlier in the context of study of charged particle dynamics on the basis of heuristic
considerations without any estimates of its accuracy. We provide a rigorous derivation of this formula and prove
that its accuracy is $O(\sqrt \varepsilon)$ (up to a logarithmic correction). This estimate for the accuracy is optimal.
Keywords:
slow-fast systems, averaging method, resonance.
Received: 27.02.2023 Accepted: 19.06.2023
Citation:
Yuyang Gao, Anatoly Neishtadt, Alexey Okunev, “On Phase at a Resonance in Slow-Fast Hamiltonian Systems”, Regul. Chaotic Dyn., 28:4-5 (2023), 585–612
Linking options:
https://www.mathnet.ru/eng/rcd1223 https://www.mathnet.ru/eng/rcd/v28/i4/p585
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Abstract page: | 53 | References: | 27 |
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