Alexey Vasiliev
Publications:
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.

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 pendulumâ€™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.

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.
