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Ukr. J. Phys. 2014, Vol. 59, N 9, p.932-938
doi: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.