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2013
Impact Factor

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:

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.  686-696
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 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.
Keywords: autoresonance, capture into resonance, adiabatic invariant, pendulum
Citation: Neishtadt A. I., Vasiliev A. A., Artemyev A. V.,  Capture into Resonance and Escape from it in a Forced Nonlinear Pendulum, Regular and Chaotic Dynamics, 2013, vol. 18, no. 6, pp. 686-696
DOI:10.1134/S1560354713060087
Neishtadt A. I., Su T.
On Phenomenon of Scattering on Resonances Associated with Discretisation of Systems with Fast Rotating Phase
2012, vol. 17, no. 3-4, pp.  359-366
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.
Keywords: systems with rotating phases, passage through a resonance, numerical integration, discretisation
Citation: Neishtadt A. I., Su T.,  On Phenomenon of Scattering on Resonances Associated with Discretisation of Systems with Fast Rotating Phase, Regular and Chaotic Dynamics, 2012, vol. 17, no. 3-4, pp. 359-366
DOI:10.1134/S1560354712030100
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, no. 4-5, pp.  564-574
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.
Keywords: surfatron acceleration, separatrix crossings, adiabatic invariant
Citation: Neishtadt A. I., Artemyev A. V., Zelenyi L. M.,  Regular and chaotic charged particle dynamics in low frequency waves and role of separatrix crossings, Regular and Chaotic Dynamics, 2010, vol. 15, no. 4-5, pp. 564-574
DOI:10.1134/S1560354710040118
Itin A. P., Neishtadt A. I.
Resonant Phenomena in Slowly Perturbed Elliptic Billiards
2003, vol. 8, no. 1, pp.  59-66
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.
Citation: Itin A. P., Neishtadt A. I.,  Resonant Phenomena in Slowly Perturbed Elliptic Billiards, Regular and Chaotic Dynamics, 2003, vol. 8, no. 1, pp. 59-66
DOI:10.1070/RD2003v008n01ABEH000225
Neishtadt A. I.
On the Accuracy of Persistence of Adiabatic Invariant in Single-frequency Systems
2000, vol. 5, no. 2, pp.  213-218
Abstract
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.
Citation: Neishtadt A. I.,  On the Accuracy of Persistence of Adiabatic Invariant in Single-frequency Systems, Regular and Chaotic Dynamics, 2000, vol. 5, no. 2, pp. 213-218
DOI:10.1070/RD2000v005n02ABEH000143
Vasiliev A. A., Itin A. P., Neishtadt A. I.
On dynamics of four globally phase-coupled oscillators with close frequencies
1998, vol. 3, no. 1, pp.  9-18
Abstract
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.
Citation: Vasiliev A. A., Itin A. P., Neishtadt A. I.,  On dynamics of four globally phase-coupled oscillators with close frequencies, Regular and Chaotic Dynamics, 1998, vol. 3, no. 1, pp. 9-18
Vasiliev A. A., Itin A. P., Neishtadt A. I.
On dynamics of four globally phase-coupled oscillators with close frequencies
1997, vol. 2, nos. 3-4, pp.  21-29
Abstract
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.
Citation: Vasiliev A. A., Itin A. P., Neishtadt A. I.,  On dynamics of four globally phase-coupled oscillators with close frequencies, Regular and Chaotic Dynamics, 1997, vol. 2, nos. 3-4, pp. 21-29
DOI:10.1070/RD1997v002n03ABEH000044

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