ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

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Pankratova E. V., Belykh V. N. Qualitative and numerical analysis of possible synchronous regimes for two inertially coupled van der Pol oscillators. Izvestiya VUZ. Applied Nonlinear Dynamics, 2011, vol. 19, iss. 4, pp. 25-39. DOI: 10.18500/0869-6632-2011-19-4-25-39

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Qualitative and numerical analysis of possible synchronous regimes for two inertially coupled van der Pol oscillators

Pankratova Evgenija Valerevna, Volga State Academy of Water Transport (VGAVT)
Belykh Vladimir Nikolaevich, Volga State Academy of Water Transport (VGAVT)

We consider a mechanical system consisting of two controlled masses that are attached to a movable platform via springs. We assume that at the absence of interaction the oscillations of both masses are described by the van der Pol equations. In this case, different modes of synchronous behavior of the masses are observed: in-phase (complete), anti-phase and phase locking. By the methods of qualitative and numerical analysis, the boundaries of the stability domains of these regimes are obtained.

  1. Huygens C. Horoloquim Oscilatorium. Apud F. Muguet, Parisiis, France;1673; English translation: The pendulum clock. Iowa State University Press, Ames; 1986.
  2. Pikovsky A, Rosenblum M, and Kurths J. Synchronization: A Universal Concept in Nonlinear Science. Cambridge: Cambridge University Press; 2001. 432 p.
  3. Korteweg DJ. Les horloges sympathiques de Huygens. Archives Neerlandaises, ser. II, tome XI. The Hague: Martinus Nijhoff; 1906. P. 273–295.
  4. Blekhman II. Synchronization in Science and Technology. New York: ASME; 1998. 255 p.
  5. Pantaleone J. Synchronization of metronomes. American Journal of Physics. 2002;70(10):992–1000. DOI: 10.1119/1.1501118.
  6. Bennett M, Schatz M, Rockwood H, and Wiesenfeld K. Huygens’s clocks. Proc. R. Soc. Lond. A. 2002;458(2019):563–579. DOI: 10.1098/rspa.2001.0888.
  7. Oud WT, Nijmeijer H, and Pogromsky AY. A study of Huijgens’ synchronization. Experimental results. In: Pettersen KY, Gravdahl JT, Nijmeijer H, editors. Group Coordination and Control. Springer; 2006. P. 191–203. DOI: 10.1007/11505532_11.
  8. Fradkov AL, Andrievsky B. Synchronization and phase relations in the motion of two-pendulum system. International Journal of Non-Linear Mechanics. 2007;42(6):895–901. DOI: 10.1016/j.ijnonlinmec.2007.03.016.
  9. Czolczynski K, Perlikovski P, Stefanski A, Kapitaniak T. Clustering and synchronization of n Huygens’ clocks. Physica A. 2009;388(24):5013–5023. DOI: 10.1016/j.physa.2009.08.033.
  10. Belykh VN, Pankratova EV, and Pogromsky AY. Two van der Pol–Duffing oscillators with Huygens coupling. In: Leonov G, Nijmeijer H, Pogromsky A, Fradkov A, editors. Dynamics and Control of Hybrid Mechanical Systems. World Scientific Publishing Co. Pte. Ltd.;2010. P. 181–194. DOI: 10.1142/9789814282321_0012.
  11. Belykh VN, Pankratova EV. Chaotic Dynamics of Two van der Pol–Duffing oscillators with Huygens coupling. Regular and Chaotic Dynamics. 2010;15(2):274–284. DOI: 10.1134/S1560354710020140.
  12. Van der Pol B. Theory of the amplitude of free and forced triode vibration. Radio Rev. 1922;1:701–710.
  13. Bautin NN. The Behavior of Dynamical Systems Near the Boundaries of the Stability Region. Mosow: Fizmatlit; 1984. 176 p. (in Russian).
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