Alexey Vasiliev
Publications:
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.
