• Українська
  • English

< | Next issue >

Current issue   Ukr. J. Phys. 2015, Vol. 59, N 9, p.932-938
https://doi.org/10.15407/ujpe59.09.0932    Paper

Danylenko V.A., Skurativskyi S.I., Skurativska I.A.

S.I. Subbotin Institute of Geophysics, Explosion Geodynamics Section, Nat. Acad. of Sci. of Ukraine (63g, Bogdan Khmelnytskyi Str., Kyiv 01054, Ukraine; e-mail: skurserg@rambler.ru)

Asymptotic Wave Solutions for the Model of a Medium with Van Der Pol Oscillators

Section: Nonlinear processes
Original Author's Text: Ukrainian

Abstract: A one-dimensional mathematical model for a complex medium with van der Pol oscillators has been studied. Using the Bogolyubov–Mitropolsky method, the wave solutions for a weakly nonlinear model are derived, with their amplitudes being described by a three-dimensional dynamical system analyzed in more details by numerical and qualitative methods. In particular, periodic, multiperiodic, and chaotic trajectories are found in the phase space of the dynamical system. Bifurcations of those regimes were considered using the Poincar´e section technique. Exact solutions are derived in the case where the three-dimensional system for amplitudes is reduced to the two-dimensional one.

Key words: nonlinear waves, van der Pol oscillator, chaotic attractor.


  1. A.S. Makarenko, Ukr. Fiz. Zh. 57, 408 (2012).
  2. M.A. Sadovskii, Vestn. Akad. Nauk SSSR 8, 3 (1986).
  3. V.A. Pal'mov, Prikl. Matem. Mech. 4, 768 (1969).
  4. L.I. Slepyan, Mir Tekhn. Tekhnol. 5, 34 (1967).
  5. V.A. Danylenko, T.B. Danevych, O.S. Makarenko, S.I. Skurativskyi, and V.A. Vladimirov, Self-Organization in Nonlocal Non-Equilibrium Media (Institute of Geophysics, Kyiv, 2011).
  6. S. Skurativskyi and I. Skurativska, http://ejta.org/en/skuratovsky1.
  7. V.A. Danylenko and S.I. Skurativskyi, Dopov. Nat. Akad. Nauk Ukr., No. 11, 108 (2008).
  8. S.I. Skurativskyi, Matem. Met. Fiz. Mech. Polya 55, No. 4, 47 (2012).
  9. V.A. Danylenko and S.I. Skurativskyi, Nonlin. Dynam. Syst. Theory 12, 365 (2012).
  10. V.A. Danylenko and S.I. Skurativskyi, Dynam. Syst. 2, 227 (2012).
  11. D.W. Storti and R.H. Rand, Int. J. Non-Linear Mech. 17, 143 (1982).
  12. A.P. Kuznetsov, N.V. Stankevich, and L.V. Tyuryukina, Izv. Vyssh. Ucheb. Zaved. Prikl. Nelin. Dinam. 16, 101 (2008).
  13. E. Camacho, R.H. Rand, and H. Howland, Int. J. Solids Struct. 41, 2133 (2004).
  14. A. Pikovskii, M. Rozenblum, and Yu. Kurts, Synchronization: A Fundamental Nonlinear Phenomenon (Teknnosfera, Moscow, 2003) (in Russian).
  15. T.A. Levanova, M.A. Komarov, and G.V. Osipov, Eur. Phys. J. Spec. Topics 222, 2417 (2013).
  16. N.N. Bogoliubov and Y.A. Mitropolsky, Asymptotic Methods in the Theory of Nonlinear Oscillations (Gordon and Breach, New York, 1961).
  17. N.V. Butenin, Yu.I. Neimark, and N.A. Fufaev, Introduction to the Theory of Nonlinear Oscillations (Nauka, Moscow, 1987) (in Russian).
  18. N.N. Bautin and E.A. Leontovich, Methods and Techniques for Qualitative Study of Dynamical Systems on the Plane (Nauka, Moscow, 1990) (in Russian).
  19. A.D. Polyanin and V.F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations (Chapman and Hall/CRC Press, Boca Raton, 2003).
  20. J. Guckenheimer and Ph. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Springer, New York, 1987).
  21. V.N. Sydorets and I.V. Pentegov, Deterministic Chaos in Nonlinear Circuits with Electric Arc (IAW, Kyiv, 2013) (in Russian).
  22. V. Palmov, Vibrations of Elasto-Plastic Bodies (Springer, Berlin, 1998).
  23. N.A. Vilchinskaya and V.N. Nikolaevskii, Fiz. Zemli 5, 91 (1984).