On phase at a resonance in slow-fast Hamiltonian systems

TL;DR

This paper rigorously derives an asymptotic formula for the phase at resonance in slow-fast Hamiltonian systems with a vanishing frequency, providing accuracy estimates and numerical validation.

Contribution

It offers a rigorous derivation and proof of the accuracy of an asymptotic phase formula at resonance in slow-fast Hamiltonian systems.

Findings

Derived an asymptotic formula for the phase at resonance.

Proved the formula's accuracy is $O(\sqrt{ ext{ε}})$ with a logarithmic correction.

Numerical results confirm the optimality of the accuracy estimate.

Abstract

We consider a slow-fast Hamiltonian system with one fast angular 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 inhomogeneous magnetic field under influence of a high-frequency electrostatic waves. Trajectories of the averaged over the fast phase system cross the resonant surface. The fast phase makes $\sim \frac {1}{\varepsilon}$ turns before arrival to the resonant surface ($\varepsilon$ is a small parameter of the problem). An asymptotic formula for the value of the phase at the arrival to 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). Numerics indicate that this estimate for the accuracy is optimal.