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

For citation:

Khrisanfova S. O., Kadina E. J., Gubina E. V., Kogan L. V., Osipov G. V. The dynamics of two nonlinearly coupled oscillators. Izvestiya VUZ. Applied Nonlinear Dynamics, 2016, vol. 24, iss. 3, pp. 4-20. DOI: 10.18500/0869-6632-2016-24-3-4-20

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
PDF icon 2016no3p004.pdf
(downloads: 197)
Language: 
Russian
Heading: 
Applied Problems of Nonlinear Oscillation and Wave Theory
Article type: 
Article
UDC: 
621.391.01
DOI: 
10.18500/0869-6632-2016-24-3-4-20

The dynamics of two nonlinearly coupled oscillators

Autors: 
Khrisanfova Svetlana Olegovna, Lobachevsky State University of Nizhny Novgorod
Kadina Elena Jurevna, Lobachevsky State University of Nizhny Novgorod
Gubina Elena Vasilevna, Lobachevsky State University of Nizhny Novgorod
Kogan Ljudmila Vladimirovna, Lobachevsky State University of Nizhny Novgorod
Osipov Grigorij Vladimirovich, Lobachevsky State University of Nizhny Novgorod
Abstract: 

In this paper the dynamics of two elastically coupled pendulums is studied. The pendulums oscillate under the influence of external rotational moments, their masses are considered to be equal. The current work is motivated by multiple applications in physics and biology that the model has. Due to the fact that most of the previous studies focused on similar systems of higher order, we believe that the current research can serve as a basis for understanding the functioning of more complex oscillatory ensembles. It is, therefore, vital to provide a complete study of the system dynamics for different parameter values. Throughout the study different regimes of the system activity are examined. Thus, non-oscillatory mode, synchronization, periodic and quasi-periodic regimes are discussed in the paper. Synchronization is often considered to be one of the most important forms of interaction between oscillatory elements of various nature. For this reason the synchronization domain is thoroughly investigated in this paper. The main results of the current research are as follow. An analytical approximation of the synchronization domain border is obtained in (d, α) parameter plane. Here d denotes the coupling strength, whereas α is the synchronization parameter. By means of numerical integration methods the approximation is also shown to be accurate. In order to provide better understanding of the regimes that exist in the system for various parameter values, bifurcation diagrams for several values of the coupling parameter in in (γ1, γ2) plane are drawn.  

Key words: 
Coupled pendulums
nonlinear coupling
synchronization
Reference: 
  1. Matrosov V.V. Dynamics of two phase locked loop system coupled through the phase discriminator // Izvestiya VUZ. Applied Nonlinear Dynamics. 2007. Vol. 15, № 3. P. 15 (in Russian).
  2. Bhansali P., Roychowdhury J. Gen-Adler: The generalized Adler’s equation for injection locking analysis in oscillators // Proceedings of the Design Automation Conference. 2009. P. 522–527.
  3. Perlikowski P., Yanchuk S., Popovych O.V., Tass P.A. Periodic patterns in a ring of delay-coupled oscillators // Phys. Rev. E. 2010. Vol. 82, № 3. P. 036208. http://link.aps.org/doi/10.1103/PhysRevE.82.036208
  4. Maistrenko Y., Penkovsky B., Rosenblum M. Solitary state at the edge of synchrony in ensembles with attractive and repulsive interactions // Phys. Rev. E. 2014.Vol. 89, № 6. P. 060901. http://link.aps.org/doi/10.1103/PhysRevE.89.060901
  5. Burylko O., Kazanovich Y., Borisyuk R. Bifurcation study of phase oscillator systems with attractive and repulsive interaction // Phys. Rev. E. 2014. Vol. 90, № 2. P. 022911. http://link.aps.org/doi/10.1103/PhysRevE.90.022911
  6. Xie J., Knobloch E., Kao Hsien-Ching. Multicluster and traveling chimera states in nonlocal phase-coupled oscillators // Phys. Rev. E. 2014. Vol. 90, № 2. P. 022919. http://link.aps.org/doi/10.1103/PhysRevE.90.022919
  7. Smirnov L.A., Kryukov A.K., Kadina E.Yu., Gubina E.V., Osipov G.V. Rotational dynamics in a pair of coupled // J. The Problems of Strength and Plasticity. 2015. Vol. 77, № 4. P. 425 (in Russian).
  8. Khibnik A.I., Braimanc Y., Kennedyd T.A.B., Wiesenfeldd K. Phase model analysis of two lasers with injected field // Physica D. 1998. Vol. 111, № 1–4. P. 295–310.
  9. Guckenheimer J., Khibnik A. Torus maps from weak coupling of strong resonances/ In book: «Methods of Qualitative Theory of Differential Equations and Related Topics» // American Mathematical Society. 2000. P. 205–218.
  10. Pikovsky A., Rosenblum M., Kurths J. Synchronization: A Universal Concept in Nonlinear Sciences / Series «The Cambridge nonlinear science series». Cambridge: Cambridge University Press, 2001. 411 p.
  11. Braun O., Kivshar Yu.S. The Frenkel–Kontorova Model: Concepts, Methods, and Applications. Berlin: Springer, 2004. P. 491.
  12. Yakushevich L.V. Nonlinear Physics of DNA. Wiley-VCH, 2004. P. 207.
  13. Leeman C., Lereh P., Racine G.A., Martinoli P. Vortex dynamics and phase transitions in a two-dimensional array of Josephson junctions // Phys. Rev. Lett. 1986. Vol. 56, № 12. P. 1291–1294.
  14. Ryu S., Yu W., Stroud D. Dynamics of an underdamped Josephson junction ladders // Phys. Rev. E. 1996. Vol. 53, № 3. P. 2190–2195.
  15. Kim B.J., Kim S., Lee S.J. Defect motions and smearing of Shapiro steps in Josephson junction ladders under magnetic frustration // Phys. Rev. B. 1995.Vol. 51, № 13. P. 8462–8466.
  16. Kim J., Choe W.G., Kim S., Lee H.J. Dynamics of Josephson junction ladders // Phys. Rev. B. 1994. Vol. 49, № 1. P. 459–464.
  17. Denniston C., Tang C. Phases of Josephson junction ladders // Phys. Rev. Lett. 1995. Vol. 75, № 21. P. 3930-3933.
  18. Qjan M., Weng J.-Z. Transitions in two sinusoidally coupled Josephson junction rotators // Annals of Physics. 2008. Vol. 323. P. 1956–1962.
  19. Fishman R.S., Stroud D. Role of long-range Coulomb interactions in granular super conductors // Phys. Rev. B. 1988. Vol. 38, № 1. P. 290–296.
  20. Yakushevich L.V., Gapa S., Awrejcewicz J. Mechanical analog of the DNA base pair-oscillations // Dynamical Systems. Theory and Applications. 2009. P. 879–886.
  21. Yakushevich L.V. Biomechanics of DNA: Rotational oscillations of bases // J. Computer Research and Modeling. 2011. Vol. 3, № 3. P. 319 (in Russian).
  22. Awrejcewicz J., Mlynarska S., Yakushevich L.V. Non-linear oscillations of DNA base pairs // J. Appl. Math. Mech. 2013. Vol. 77, № 4. P. 392 (in Russian).
  23. Krueger A., Protozanova E., Frank-Kamenetskii M. Sequence-dependent basepair opening in DNA double helix // Biophys. J. 2006. Vol. 90. P. 3091–3099.
  24. Takeno S., Peyrard M. Nonlinear modes in coupled rotator models // Physica D. 1996. Vol. 92. P. 140–163.
  25. Zhang F. Kink shape modes and resonant dynamics in sine-lattices // Physica D. 1997. Vol. 110. P. 51–61.
  26. Kosterlitz J.M., Thouless D.J. Ordering, metastability and phase transitions in twodimensional systems // J. Phys. C. Solid State Phys. 1973. Vol. 6. P. 1181–1203.
  27. Antoni M., Ruffo S. Clustering and relaxation in Hamiltonian long-range dynamics // Phys. Rev. E. 1995. Vol. 52, № 3. P. 2361–2374.
  28. Wang X.Y., Taylor P.L. Devil’s staircase, critical thickness, and propagating fingers in antiferroelectric liquid crystals // Phys. Rev. Lett. 1996. Vol. 76, № 4. P. 640–643.
  29. Fillaux F., Carlile C.J. Inelastic-neutron-scattering study of methyl tunneling and the quantum sine-Gordon breather in isotopic mixtures of 4-methyl-pyridine at low temperature // Phys. Rev. B. 1990. Vol. 42, №10. P. 5990–6006.
  30. Zhang F., Collins M.A., Kivshar Yu.S. Kinks and conformational defects in nonlinear chains // Phys. Rev. E. 1995. Vol. 51, № 4. P. 3774.
  31. Fillaux F., Carlile C.J., Kearley G.J. Inelastic-neutron-scattering study at low temperature of the quantum sine-Gordon breather in 4-methyl-pyridine with partially deuterated methyl groups // Phys. Rev. B. 1991. Vol. 44, № 22. P. 12280–12293.
  32. Adler R. A study of locking phenomena in oscillators // Proceedings of the IRE.1946. Vol. 34, № 6. P. 351–357.
  33. Osipov G., Kurths J., Zhou Ch. Synchronization in Oscillatory Networks. Berlin:Springer, 2007.
Received: 
14.04.2016
Accepted: 
17.05.2016
Published: 
30.06.2016
Short text (in English):
PDF icon a2016no3p004.pdf
(downloads: 105)
  • 2151 reads