Anatoly Neishtadt

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.

Slow-fast dynamics and resonant phenomena can be found in a wide range of physical systems, including problems of celestial mechanics, fluid mechanics, and charged particle dynamics. Important resonant effects that control transport in the phase space in such systems are resonant scatterings and trappings. For systems with weak diffusive scatterings the transport properties can be described with the Chirikov standard map, and the map parameters control the transition between stochastic and regular dynamics. In this paper we put forward the map for resonant systems with strong scatterings that result in nondiffusive drift in the phase space, and trappings that produce fast jumps in the phase space. We demonstrate that this map describes the transition between stochastic and regular dynamics and find the critical parameter values for this transition.

We consider the 2D Schrödinger equation with variable potential in the narrow domain diffeomorphic to the wedge with the Dirichlet boundary condition. The corresponding classical problem is the billiard in this domain. In general, the corresponding dynamical system is not integrable. The small angle is a small parameter which allows one to make the averaging and reduce the classical dynamical system to an integrable one modulo exponential small correction. We use the quantum adiabatic approximation (operator separation of variables) to construct the asymptotic eigenfunctions (quasi-modes) of the Schr¨odinger operator. We discuss the relation between classical averaging and constructed quasi-modes. The behavior of quasi-modes in the neighborhood of the cusp is studied. We also discuss the relation between Bessel and Airy functions that follows from different representations of asymptotics near the cusp.

We study the dynamics of a nonlinear pendulum under a periodic force with small amplitude and slowly decreasing frequency. It is well known that when the frequency of the external force passes through the value of the frequency of the unperturbed pendulum’s oscillations, the pendulum can be captured into resonance. The captured pendulum oscillates in such a way that the resonance is preserved, and the amplitude of the oscillations accordingly grows. We consider this problem in the frames of a standard Hamiltonian approach to resonant phenomena in slow-fast Hamiltonian systems developed earlier, and evaluate the probability of capture into resonance. If the system passes through resonance at small enough initial amplitudes of the pendulum, the capture occurs with necessity (so-called autoresonance). In general, the probability of capture varies between one and zero, depending on the initial amplitude. We demonstrate that a pendulum captured at small values of its amplitude escapes from resonance in the domain of oscillations close to the separatrix of the pendulum, and evaluate the amplitude of the oscillations at the escape.

Numerical integration of ODEs by standard numerical methods reduces continuous time problems to discrete time problems. Discrete time problems have intrinsic properties that are absent in continuous time problems. As a result, numerical solution of an ODE may demonstrate dynamical phenomena that are absent in the original ODE. We show that numerical integration of systems with one fast rotating phase leads to a situation of such kind: numerical solution demonstrates phenomenon of scattering on resonances that is absent in the original system.

We consider interaction of charged particles with an electromagnetic (electrostatic) low frequency wave propagating perpendicular to a uniform background magnetic field. The effects of particle trapping by the wave and further acceleration of a surfatron type are discussed in details. Method for this analysis based on the adiabatic theory of separatrix crossing is used. It is shown that particle can unlimitedly accelerate in the trapping in electromagnetic waves and energy of particle does not increase for the system with electrostatic wave.

We consider an elliptic billiard whose shape slowly changes. During slow evolution of the billiard certain resonance conditions can be fulfilled. We study the phenomena of capture into a resonance and scattering on a resonance which lead to the destruction of the adiabatic invariance in the system.

A modified method of A.A. Slutskin (1963) of analytical extension to the complex time plane of solutions of a single-frequency nonlinear Hamiltonian system with slowly varying parameters is considered. On the basis of this method a proof of the estimate for the accuracy of persistence of adiabatic invariant due to A.A. Slutskin is given for such systems.

An array of four globally phase-coupled oscillators with slightly different eigenfrequencies is considered. In the case of equal frequencies the system is reduced to integrable, with almost all phase closed trajectories. In the case of different but close to each other eigenfrequencies the system is treated with the use of the averaging method. It is shown that probabilistic phenomena take place in the system: at separatrices of the unperturoed problem the phase flow splits itself quasi-randomly between various regions of the phase space. Formulas are obtained, describing probabilities of capture into various regions.

An array of four globally phase-coupled oscillators with slightly different eigenfrequencies is considered. In the case of equal frequencies the system is reduced to an integrable one with almost all phase trajectories being closed. In the case of different but close to each other eigenfrequencies the system is treated with the use of the averaging method. It is shown that probabilistic phenomena take place in the system: the phase flow is divided quasi-randomly between various regions of the phase space when passing through separatrices of the unperturbed problem. Formulas describing probabilities of the phase point transition to different regions are obtained.