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Regular and Chaotic Dynamics, 2023, Volume 28, Issue 4-5, Pages 585–612
DOI: https://doi.org/10.1134/S1560354723040068
(Mi rcd1223)
 

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
References:
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.
Funding agency Grant number
Leverhulme Trust RPG-2018-143
The work was supported by the Leverhulme Trust (Grant No. RPG-2018-143).
Received: 27.02.2023
Accepted: 19.06.2023
Document Type: Article
MSC: 34C29
Language: English
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
Citation in format AMSBIB
\Bibitem{GaoNeiOku23}
\by Yuyang Gao, Anatoly Neishtadt, Alexey Okunev
\paper On Phase at a Resonance in Slow-Fast Hamiltonian Systems
\jour Regul. Chaotic Dyn.
\yr 2023
\vol 28
\issue 4-5
\pages 585--612
\mathnet{http://mi.mathnet.ru/rcd1223}
\crossref{https://doi.org/10.1134/S1560354723040068}
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