Anatoly Neishtadt
Profsoyuznaya St. 84/32, Moscow, 117810 Russia; Loughborough, Leicestershire LE11 3TU, UK
Space Research Institute, RAS; Dept. of Math. Sciences, Loughborough University
Publications:
Gao Y., Neishtadt A. I., Okunev A.
On Phase at a Resonance in SlowFast Hamiltonian Systems
2023, vol. 28, nos. 45, pp. 585612
Abstract
We consider a slowfast 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 highfrequency 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.

Artemyev A. V., Neishtadt A. I., Vasiliev A. A.
A Map for Systems with Resonant Trappings and Scatterings
2020, vol. 25, no. 1, pp. 210
Abstract
Slowfast 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.

Dobrokhotov S. Y., Minenkov D. S., Neishtadt A. I., Shlosman S. B.
Classical and Quantum Dynamics of a Particle in a Narrow Angle
2019, vol. 24, no. 6, pp. 704716
Abstract
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 (quasimodes) of the Schr¨odinger operator. We discuss the relation
between classical averaging and constructed quasimodes. The behavior of quasimodes 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.

Neishtadt A. I., Vasiliev A. A., Artemyev A. V.
Capture into Resonance and Escape from it in a Forced Nonlinear Pendulum
2013, vol. 18, no. 6, pp. 686696
Abstract
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 slowfast 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 (socalled 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.

Neishtadt A. I., Su T.
On Phenomenon of Scattering on Resonances Associated with Discretisation of Systems with Fast Rotating Phase
2012, vol. 17, nos. 34, pp. 359366
Abstract
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.

Neishtadt A. I., Artemyev A. V., Zelenyi L. M.
Regular and chaotic charged particle dynamics in low frequency waves and role of separatrix crossings
2010, vol. 15, nos. 45, pp. 564574
Abstract
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.

Itin A. P., Neishtadt A. I.
Resonant Phenomena in Slowly Perturbed Elliptic Billiards
2003, vol. 8, no. 1, pp. 5966
Abstract
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.

Neishtadt A. I.
On the Accuracy of Persistence of Adiabatic Invariant in Singlefrequency Systems
2000, vol. 5, no. 2, pp. 213218
Abstract
A modified method of A.A. Slutskin (1963) of analytical extension to the complex time plane of solutions of a singlefrequency 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.

Vasiliev A. A., Itin A. P., Neishtadt A. I.
On dynamics of four globally phasecoupled oscillators with close frequencies
1998, vol. 3, no. 1, pp. 918
Abstract
An array of four globally phasecoupled 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 quasirandomly between various regions of the phase space. Formulas are obtained, describing probabilities of capture into various regions.

Vasiliev A. A., Itin A. P., Neishtadt A. I.
On dynamics of four globally phasecoupled oscillators with close frequencies
1997, vol. 2, nos. 34, pp. 2129
Abstract
An array of four globally phasecoupled 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 quasirandomly 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.
