QUALITATIVE AND NUMERICAL ANALYSIS OF POSSIBLE SYNCHRONOUS REGIMES FOR TWO INERTIALLY COUPLED VAN DER POL OSCILLATORS
Cite this article as:
Pankratova Е. 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: https://doi.org/10.18500/0869-6632-2011-19-4-25-39
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.
3. Korteweg D.J. Les horloges sympathiques de Huygens. Archives Neerlandaises, ser. II, tome XI, pp. 273-295. The Hague: Martinus Nijhoff, 1906.
4. Blekhman I.I. Synchronization in science and technology. New York: ASME, 1998.
5. Pantaleone J. Synchronization of metronomes // American Journal of Physics. 2002. Vol. 70, No 10. P. 992.
6. Bennett M., Schatz M., Rockwood H. and Wiesenfeld K. Huygens’s clocks // Proc. R. Soc. Lond. A. 2002. Vol. 458. 2019. P. 563.
7. Oud W.T., Nijmeijer H. and Pogromsky A.Yu. A study of Huijgens’ synchronization. Experimental results // Group Coordinations and Control / K.Y. Pettersen, J.T. Gravdahl, H. Nijmeijer (eds). Springer, 2006.
8. Fradkov A.L., Andrievsky B. Synchronization and phase relations in the motion of two-pendulum system // International Journal of Non-Linear Mechanics. 2007. Vol. 42, No 6. P. 895.
9. Czolczynski K., Perlikovski P., Stefanski A., Kapitaniak T. Clustering and synchronization of n Huygens’ clocks // Physica A. 2009. Vol. 388. P. 5013.
10. Belykh V.N., Pankratova E.V. and Pogromsky A.Y. Two van der Pol–Duffing oscillators with Huygens coupling // Dynamics and Control of Hybrid Mechanical Systems / Ed. by G. Leonov, H. Nijmeijer, A. Pogromsky and A. Fradkov. World Scientific Publishing Co. Pte. Ltd. P. 181. 2010.
11. Belykh V.N., Pankratova E.V. Chaotic Dynamics of Two van der Pol–Duffing oscillators with Huygens coupling // Regular and Chaotic Dynamics. 2010. Vol. 15, No 2. P. 274.
12. Van der Pol B. Theory of the amplitude of free and forced triod vibration // Radio Rev. 1922. Vol. 1. P. 701.
13. Баутин Н.Н. Поведение динамических систем вблизи границ области устойчивости. М.: Наука. Главная редакция физико-математической литературы, 1984.
BibTeX
author = {Е. V. Pankratova and V. N. Belykh},
title = {QUALITATIVE AND NUMERICAL ANALYSIS OF POSSIBLE SYNCHRONOUS REGIMES FOR TWO INERTIALLY COUPLED VAN DER POL OSCILLATORS},
year = {2011},
journal = {Izvestiya VUZ. Applied Nonlinear Dynamics},
volume = {19},number = {4},
url = {https://old-andjournal.sgu.ru/en/articles/qualitative-and-numerical-analysis-of-possible-synchronous-regimes-for-two-inertially},
address = {Саратов},
language = {russian},
doi = {10.18500/0869-6632-2011-19-4-25-39},pages = {25--39},issn = {0869-6632},
keywords = {Sinchronization,attractors,van der Pol equations,control input.},
abstract = {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. }}