Abstract:
The mechanism of formation of multistability in a van der Pol generator with time-delayed feedback is studied. It is found that the developed multistability is formed through a sequence of two types of bifurcations: supercritical Andronov–Hopf bifurcation and subcritical Neimark–Sacker bifurcation. With variation of the control parameters, the fixed point repeatedly undergoes supercritical Andronov–Hopf bifurcation, which leads to an increase in the number of saddle cycles. The limit cycles acquire stability after a number of subcritical Neimark– Sacker bifurcations. The dynamics of the system is studied in a wide range of control parameters values.
Keywords:
Time delay, bifurcations, multistability.
Citation:
B. I. Maksim, N. M. Ryskin, “Bifurcational mechanism of formation of developed multistability in a van der Pol oscillator with time-delayed feedback”, Nelin. Dinam., 13:2 (2017), 151–164
\Bibitem{BalRys17}
\by B.~I.~Maksim, N.~M.~Ryskin
\paper Bifurcational mechanism of formation of developed multistability in a van der Pol oscillator with time-delayed feedback
\jour Nelin. Dinam.
\yr 2017
\vol 13
\issue 2
\pages 151--164
\mathnet{http://mi.mathnet.ru/nd557}
\crossref{https://doi.org/10.20537/nd1702001}
\elib{https://elibrary.ru/item.asp?id=29443374}
Linking options:
https://www.mathnet.ru/eng/nd557
https://www.mathnet.ru/eng/nd/v13/i2/p151
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A. B. Adilova, M. I. Balakin, S. A. Gerasimova, N. M. Ryskin, “Bifurcation analysis of multistability of synchronous states in the system of two delay-coupled oscillators”, Chaos, 31:11 (2021), 113103
A. B. Adilova, N. M. Ryskin, “Influence of the delay on mutual synchronization of two coupled gyrotrons”, Radiophys. Quantum Electron., 63:9-10 (2021), 703–715
V. Ermolaev, Yu. Kropotov, A. Proskuryakov, “Modeling of neural systems and networks by functional differential equations”, 2020 Vi International Conference on Information Technology and Nanotechnology (IEEE Itnt-2020), ed. D. Kudryashov, IEEE, 2020
I. S. Kashchenko, S. A. Kashchenko, “Dynamics of equation with two delays modelling the number of population”, Izv. vuzov. Prikladnaya nelineinaya dinamika, 27:2 (2019), 21–38
M. Balakin, A. Gulyaev, A. Kazaryan, O. Yarovoy, “Synchronization of oscillations in coupled multimode optoelectronic oscillators: bifurcation analysis”, Saratov Fall Meeting 2017: Laser Physics and Photonics XVIII; and Computational Biophysics and Analysis of Biomedical Data IV, Proceedings of SPIE, 10717, eds. V. Derbov, D. Postnov, SPIE-Int. Soc. Optical Engineering, 2018, 107172H
N. Stankevich, E. Volkov, “Multistability in a three-dimensional oscillator: tori, resonant cycles and chaos”, Nonlinear Dyn., 94:4 (2018), 2455–2467
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