Repulsion from resonances (Q2900340)

The subject of consideration is the slow-fast system NEWLINE\[NEWLINE\begin{aligned} \dot I &= \alpha_1(I, \phi, \theta)+\varepsilon\alpha_2(I, \phi, \theta, \varepsilon), \\ \dot \phi &= p(I)+\beta_1(I, \phi, \theta) + \varepsilon \beta_2(I, \phi, \theta), \\ \dot \theta &=\varepsilon^{-1}\omega(I, \phi) + \eta (I, \phi, \theta, \varepsilon)\end{aligned} NEWLINE\]NEWLINE with periodic fast motion and integrable slow motion in the presence of both weak and strong resonances. Here \(\alpha_1\) and \(\beta_1\) satisfy NEWLINE\[NEWLINE \int_0^1 \alpha_1(I, \phi, \theta)\, \text{d}\theta = \int_0^1 \beta_1(I, \phi, \theta)\, \text{d}\theta = 0, NEWLINE\]NEWLINE \(I\) varies over an interval \([I_1, I_2]\) and \(\phi, \theta\) vary over the circle \(\mathbb{R}/\mathbb{Z}\).NEWLINENEWLINEThe average principle guarantees that away from resonant surface \(\{\omega=0\}\) the effective dynamics of slow variables is given by the averaged equation NEWLINE\[NEWLINE \dot{\overline{I}}=0, \quad \dot{\overline{\phi}}=p(\overline{I}). NEWLINE\]NEWLINE In particular \(I\) is an adiabatic invariant.NEWLINENEWLINEAssuming that the initial phases \(\phi(0), \theta(0)\) are random and that appropriate non-degeneracy assumptions are satisfied the author proves that the effective evolution of the adiabatic invariant on a longer time scale is given by a Markov process. This Markov process consists of the motion along the trajectories of a vector field with occasional jumps. The generator of the limiting process is computed from the dynamics of the system near strong resonance.NEWLINENEWLINETo illustrate the main results, the parameters of the limiting process are computed for a particle moving in a rapidly oscillating force field and for a particle moving on a narrow cylinder in the presence of a magnetic field.