@article{GlebovKiselevTarkhanov2010,
  author    = {Glebov, Sergei and Kiselev, Oleg and Tarkhanov, Nikolai Nikolaevich},
  title     = {Autoresonance in a dissipative system},
  issn      = {1751-8113},
  doi       = {10.1088/1751-8113/43/21/215203},
  year      = {2010},
  abstract  = {We study the autoresonant solution of Duffing's equation in the presence of dissipation. This solution is proved to be an attracting set. We evaluate the maximal amplitude of the autoresonant solution and the time of transition from autoresonant growth of the amplitude to the mode of fast oscillations. Analytical results are illustrated by numerical simulations.},
  language  = {en}
}
@article{GlebovKiselevTarkhanov2010,
  author    = {Glebov, Sergei and Kiselev, Oleg and Tarkhanov, Nikolai Nikolaevich},
  title     = {Weakly nonlinear dispersive waves under parametric resonance perturbation},
  issn      = {0022-2526},
  doi       = {10.1111/j.1467-9590.2009.00460.x},
  year      = {2010},
  abstract  = {We consider a solution of the nonlinear Klein-Gordon equation perturbed by a parametric driver. The frequency of parametric perturbation varies slowly and passes through a resonant value, which leads to a solution change. We obtain a new connection formula for the asymptotic solution before and after the resonance.},
  language  = {en}
}
@article{GlebovKiselevTarkhanov2011,
  author    = {Glebov, Sergei and Kiselev, Oleg and Tarkhanov, Nikolai Nikolaevich},
  title     = {Forced nonlinear resonance in a system of coupled oscillators},
  series = {Chaos : an interdisciplinary journal of nonlinear science},
  volume    = {21},
  journal   = {Chaos : an interdisciplinary journal of nonlinear science},
  number    = {2},
  publisher = {American Institute of Physics},
  address   = {Melville},
  issn      = {1054-1500},
  doi       = {10.1063/1.3578047},
  pages     = {7},
  year      = {2011},
  abstract  = {We consider a resonantly perturbed system of coupled nonlinear oscillators with small dissipation and outer periodic perturbation. We show that for the large time t similar to s(-2) one component of the system is described for the most part by the inhomogeneous Mathieu equation while the other component represents pulsation of large amplitude. A Hamiltonian system is obtained which describes for the most part the behavior of the envelope in a special case. The analytic results agree with numerical simulations.},
  language  = {en}
}
@article{KiselevTarkhanov2014,
  author    = {Kiselev, Oleg and Tarkhanov, Nikolai Nikolaevich},
  title     = {Scattering of trajectories at a separatrix under autoresonance},
  series = {Journal of mathematical physics},
  volume    = {55},
  journal   = {Journal of mathematical physics},
  number    = {6},
  publisher = {American Institute of Physics},
  address   = {Melville},
  issn      = {0022-2488},
  doi       = {10.1063/1.4875105},
  pages     = {24},
  year      = {2014},
  abstract  = {The subject of this paper is solutions of an autoresonance equation. We look for a connection between the parameters of the solution bounded as t -> -infinity, and the parameters of two two-parameter families of solutions as t -> infinity. One family consists of the solutions which are not captured into resonance, and another of those increasing solutions which are captured into resonance. In this way we describe the transition through the separatrix for equations with slowly varying parameters and get an estimate for parameters before the resonance of those solutions which may be captured into autoresonance. (C) 2014 AIP Publishing LLC.},
  language  = {en}
}
@unpublished{KiselevTarkhanov2013,
  author    = {Kiselev, Oleg and Tarkhanov, Nikolai Nikolaevich},
  title     = {The capture of a particle into resonance at potential hole with dissipative perturbation},
  issn      = {2193-6943},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-64725},
  year      = {2013},
  abstract  = {We study the capture of a particle into resonance at a potential hole with dissipative perturbation and periodic outside force. The measure of resonance solutions is evaluated. We also derive an asymptotic formula for the parameter range of those solutions which are captured into resonance.},
  language  = {en}
}
@unpublished{AntonioukKiselevStepanenkoetal.2012,
  author    = {Antoniouk, Alexandra Viktorivna and Kiselev, Oleg and Stepanenko, Vitaly and Tarkhanov, Nikolai Nikolaevich},
  title     = {Asymptotic solutions of the Dirichlet problem for the heat equation at a characteristic point},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-61987},
  year      = {2012},
  abstract  = {The Dirichlet problem for the heat equation in a bounded domain is characteristic, for there are boundary points at which the boundary touches a characteristic hyperplane t = c, c being a constant. It was I.G. Petrovskii (1934) who first found necessary and sufficient conditions on the boundary which guarantee that the solution is continuous up to the characteristic point, provided that the Dirichlet data are continuous. This paper initiated standing interest in studying general boundary value problems for parabolic equations in bounded domains. We contribute to the study by constructing a formal solution of the Dirichlet problem for the heat equation in a neighbourhood of a characteristic boundary point and showing its asymptotic character.},
  language  = {en}
}