Oscillations Under Hysteretic Conditions: From Simple Oscillator to Discrete Sine-Gordon Model

  • Mikhail E. SemenovEmail author
  • Olga O. Reshetova
  • Akim V. Tolkachev
  • Andrey M. Solovyov
  • Peter A. Meleshenko
Conference paper
Part of the Springer Proceedings in Physics book series (SPPHY, volume 228)


In this paper we study the resonance properties of oscillating system in the case when the energy pumping is made by external source of hysteretic nature. We investigate the unbounded solutions of autonomous oscillating system with hysteretic block with a negative spin. The influence of a hysteretic block on an oscillator in the presence of Coulomb and viscous friction is also investigated. Namely, we establish the appearance of self-oscillating regimes for both kinds of friction. A separate part of this work is devoted to synchronization of periodic self-oscillations by a harmonic external force. Using the small parameter approach it is shown that the width of “trapping” band depends on the intensity (amplitude) of the external impact. Also in this work we introduce the novel class of hysteretic operators with random parameters. We consider the definition of these operators in terms of the “input-output” relations, namely: for all permissible continuous inputs corresponds the output in the form of stochastic Markovian process. The properties of such operators are also considered and discussed on the example of a non-ideal relay with random parameters. Application of hysteretic operator with stochastic parameters is demonstrated on the example of simple oscillating system and the results of numerical simulations are presented. We consider also a nonlinear dynamical system which is a set of nonlinear oscillators coupled by springs with hysteretic blocks (modified sine-Gordon system or hysteretic sine-Gordon model where the hysteretic nonlinearity is simulated by the Bouc-Wen model). We investigate the wave processes (namely, the solitonic solutions) in such a system taking into account the hysteretic nonlinearity in the coupling.


Hysteresis Oscillator Non-ideal relay Random parameters Sine-gordon model Solitonic solutions Bouc-wen model 



The works of authors (Introduction, Oscillator under hysteretic force and Oscillator under force with random parameters (Sects. 12.112.3)) was supported by the RFBR (Grants 17-01-00251-a, 18-08-00053-a, and 19-08-00158-a). The work of M.E. Semenov and P.A. Meleshenko (Hysteresis in discrete sine-Gordon model (Sect. 12.4)) was supported by the RSF grant No. 19-11-0197.


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Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Mikhail E. Semenov
    • 1
    • 2
    • 3
    • 4
    Email author
  • Olga O. Reshetova
    • 5
  • Akim V. Tolkachev
    • 6
  • Andrey M. Solovyov
    • 5
  • Peter A. Meleshenko
    • 7
  1. 1.Geophysical Survey of Russia Academy of SciencesObninskRussia
  2. 2.Meteorology DepartmentZhukovsky–Gagarin Air Force AcademyVoronezhRussia
  3. 3.Digital Technologies DepartmentVoronezh State UniversityVoronezhRussia
  4. 4.Mathematics DepartmentVoronezh State University of Architecture and Civil EngineeringVoronezhRussia
  5. 5.Digital Technologies DepartmentVoronezh State UniversityVoronezhRussia
  6. 6.Zhukovsky–Gagarin Air Force AcademyVoronezhRussia
  7. 7.Communication DepartmentZhukovsky–Gagarin Air Force AcademyVoronezhRussia

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