Skip to main content
SpringerLink
Log in

Stability of the synchronization manifold in nearest neighbor nonidentical van der Pol-like oscillators

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

We investigate the stability of the synchronization manifold in a ring and in an open-ended chain of nearest neighbor coupled self-sustained systems, each self-sustained system consisting of multi-limit cycle van der Pol oscillators. Such a model represents, for instance, coherent oscillations in biological systems through the case of an enzymatic-substrate reaction with ferroelectric behavior in a brain waves model. The ring and open-ended chain of identical and nonidentical oscillators are considered separately. By using the Master Stability Function approach (for the identical case) and the complex Kuramoto order parameter (for the nonidentical case), we derive the stability boundaries of the synchronized manifold. We have found that synchronization occurs in a system of many coupled modified van der Pol oscillators, and it is stable even in the presence of a spread of parameters.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Pikovsky, A., Rosemblum, M., Kurths, J.: Synchronization: A Universal Concept in Nonlinear Systems. Cambridge University Press, Cambridge (2001)

    MATH  Google Scholar 

  2. Boccaletti, S., Kurths, J., Valladares, D.L., Osipov, G., Zhou, C.S.: The synchronization of chaotic systems. Phys. Rep. 366(1), 1–101 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  3. Manrubia, S.C., Mikhailov, A.S., Zanette, A.H.: Emergence of Dynamical Order. World Scientific, Singapore (2004)

    MATH  Google Scholar 

  4. Pecora, L.M., Carroll, T.L.: Master stability functions for synchronized coupled systems. Phys. Rev. Lett. 80, 2109–2112 (1998)

    Article  Google Scholar 

  5. Barahona, M., Pecora, L.M.: Synchronization in small-world systems. Phys. Rev. Lett. 89, 054101 (2002)

    Article  Google Scholar 

  6. Hu, G., Yang, J., Liu, W.: Instability and controllability of linearly coupled oscillators: eigenvalue analysis. Phys. Rev. E 58, 4440–4453 (1998)

    Article  Google Scholar 

  7. Zhan, M., Hu, G., Yang, J.: Synchronization of chaos in coupled systems. Phys. Rev. E 62, 2963–2966 (2000)

    Article  Google Scholar 

  8. Chen, Y., Rangarajan, G., Ding, M.: General stability analysis of synchronized dynamics in coupled systems. Phys. Rev. E 67, 026209 (2003)

    Article  Google Scholar 

  9. Kaiser, F.: Coherent oscillations in biological systems: Interaction with extremely low frequency fields. Radio Sci. 17(5S), 17S–22S (1982)

    Article  Google Scholar 

  10. Decroley, O., Goldbeter, A.: Birhythmicity, chaos, and other patterns of temporal self-organization in a multiply regulated biochemical system. Proc. Natl. Acad. Sci. U.S.A. 79, 6917–6921 (1982)

    Article  Google Scholar 

  11. Goldbeter, A.: Biochemical Oscillations and Cellular Rhythms. The Molecular Bases of Periodic and Chaotic Behaviour. Cambridge University Press, Cambridge (1996)

    Book  MATH  Google Scholar 

  12. Winfree, A.T.: The Geometry of Biological Time. Springer, New York (2001)

    MATH  Google Scholar 

  13. Murray, J.D.: Mathematical Biology. Springer, Berlin (1993)

    Book  MATH  Google Scholar 

  14. Goldbeter, A.: Computational approaches to cellular rhythms. Nature 420, 238–245 (2002)

    Article  Google Scholar 

  15. Kaiser, F., Eichwald, C.: Bifurcation Structure of a Driven multiple-limit-cycle Van der Pol oscillator (I): the superharmonic resonance structure. Int. J. Bifurc. Chaos 1, 485–491 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  16. Eichwald, C., Kaiser, F.: Bifurcation structure of a driven multiple-limit-cycle Van der Pol oscillator (II): symmetry-breaking crisis and intermittency. Int. J. Bifurc. Chaos 1, 711–715 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  17. Choi, J.D., Hwang, C.J.: An interaction interface for multiple agents on shared 3D display. In: Luo, Y. (ed.) CDVE. LNCS, vol. 3675, pp. 71–78. Springer, Berlin (2005)

    Google Scholar 

  18. Wiesenfeld, K., Colet, P., Strogatz, S.H.: Synchronization transitions in a disordered Josephson series array. Phys. Rev. Lett. 76, 404–407 (1996)

    Article  Google Scholar 

  19. Wiesenfeld, K., Colet, P., Strogatz, S.H.: Frequency locking in Josephson arrays: connection with the Kuramoto model. Phys. Rev. E 57, 1563–1569 (1998)

    Article  Google Scholar 

  20. Barbara, P., Cawthorne, A.B., Shitov, S.V., Lobb, C.J.: Stimulated emission and amplification in Josephson junction arrays. Phys. Rev. Lett. 82, 1963–1966 (1999)

    Article  Google Scholar 

  21. Schenato, L., Songhwai, O.H., Sastry, S., Bose, P.: Swarm coordination for pursuit evasion games using sensor networks, robotics and automation, ICRA. In: Proceedings of the 2005 IEEE International Conference, vol. 18, no. 22, p. 2493 (2005)

  22. Richard Ivry, B., Richardson, C.T.: Temporal control and coordination: the multiple timer model. Brain Cogn. 48, 117 (2002)

    Article  Google Scholar 

  23. Michael Rich, W.: Heart failure in the 21st century: a cardiogeriatric syndrome. J. Gerontol., Med. Sci. A 56(2), M88–M96 (2001)

    Google Scholar 

  24. Enjieu Kadji, H.G., Chabi Orou, J.B., Yamapi, R., Woafo, P.: Nonlinear dynamics and strange attractor in the biological system. Chaos Solitons Fractals 32, 862–882 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  25. Enjieu Kadji, H.G., Yamapi, R., Chabi Orou, J.B.: Synchronization of two coupled self-excited systems with multi-limit cycles. Chaos 17, 033113 (2007)

    Article  MathSciNet  Google Scholar 

  26. Yamapi, R., Nana Nbendjo, B.R., Enjieu Kadji, H.G.: Dynamics and active control of motion of a driven multi-limit-cycle van der Pol oscillator. Int. J. Bifurc. Chaos 17(4), 1343–1354 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  27. Nana, B., Woafo, P.: Synchronization in a ring of four mutually coupled van der Pol oscillators: theory and experiment. Phys. Rev. E 74, 046213 (2006)

    Article  Google Scholar 

  28. Kuznetov, A.P., Roman, J.P.: Properties of synchronization in the systems of non-identical coupled van der Pol and van der Pol-Duffing oscillators. Broadband synchronization. Physica D 238, 1499–1509 (2009)

    Article  MathSciNet  Google Scholar 

  29. Barròn, M.A., Sehn, M.: Synchronization of four coupled van der Pol oscillators. Nonlinear Dyn. 56, 357–367 (2009)

    Article  Google Scholar 

  30. Kamoun, P., Lavoine, A.H., Verneuil de, H.: Biochimie et Biologie Moleculaire. Flammarion, Paris (2003)

    Google Scholar 

  31. Fukui, K., Nogi, S.: Power combining ladder network with many active devices. IEEE Trans. Microwave Theor. Tech. 28, 1059–1067 (1980)

    Article  MathSciNet  Google Scholar 

  32. Fukui, K., Nogi, S.: Mode analytical study of cylindrical cavity power combiners. IEEE Trans. Microwave Theor. Tech. 34, 943–951 (1986)

    Article  Google Scholar 

  33. Appelbe, B.: Existence of multiple cycles in a van der Pol system with hysteresis in the inductance. J. Phys., Conf. Ser. 55, 1–11 (2006)

    Article  Google Scholar 

  34. Boccaletti, S., Latorab, V., Moreno, Y., Chavez, M., Hwang, D.-U.: Complex networks: structure and dynamics. Phys. Rep. 424, 175–308 (2006)

    Article  MathSciNet  Google Scholar 

  35. Pecora, L.M.: Synchronization conditions and desynchronizing patterns in coupled limit-cycle and chaotic systems. Phys. Rev. E 58(1), 347–360 (1998)

    Article  MathSciNet  Google Scholar 

  36. Tsaneva-Atanasova, K., Yule, D.I., Sneyd, J.: Calcium oscillations in a triplet of pancreatic acinar cells. Biophys. J 88, 1535–1551 (2005)

    Article  Google Scholar 

  37. Strogatz, S.H.: Exploring complex networks. Nature 410, 268–276 (2001)

    Article  Google Scholar 

  38. Kaiser, F.: Coherent modes in biological systems. In: Illinger K.H. (ed.) Biological Effects of Nonionizing Radiation. A.C.S Symp. Series, vol. 157 (1981)

  39. Fröhlich, H.: Long-range coherence and energy storage in biological systems. Int. J. Quant. Chem. 2(5), 641–649 (1968)

    Article  Google Scholar 

  40. Fröhlich, H.: Quantum mechanical concepts in biology, in theoretical physics and biology, Marois (ed.), vol. 13 (1969)

  41. Kuramoto, Y.: Chemical Oscillations, Waves and Turbulence. Springer, Berlin (1984)

    MATH  Google Scholar 

  42. Acebrón, J.A., Bonilla, I.L., Perez Vicente, C.J., Ritrot, F., Spigler, R.: The Kuramoto model: a simple paradigm for synchronization phenomena. Rev. Mod. Phys. 77, 137–185 (2005)

    Article  Google Scholar 

  43. Arenasa, A., Daz-Guilera, A., Kurthsd, J., Moreno, Y., Zhou, C.: Synchronization in complex networks. Phys. Rep. 469, 93–153 (2008)

    Article  MathSciNet  Google Scholar 

  44. Peles, S., Wiesenfeld, K.: Synchronization law for a van der Pol array. Phys. Rev. E 68, 026220.1–026220.8 (2003)

    Article  Google Scholar 

  45. Brusselbach, H., Cris Jones, D., Mangir, M.S., Minden, M., Rogers, J.L.: Self-organized coherence in fiber laser arrays. Opt. Lett. 30, 1339–1341 (2005)

    Article  Google Scholar 

  46. Strogatz, S.H., Abrams, D.M., Mcrobie, A., Eckhardt, B., Ott, E.: Theoretical mechanics: crowd synchrony on the millennium bridge. Nature 438, 43–44 (2005)

    Article  Google Scholar 

  47. Daniels, B.C., Dissanayake, S.T., Trees, B.R.: Synchronization of coupled rotators: Josephson junction ladders and the locally coupled Kuramoto model. Phys. Rev. E 67, 026216 (2003)

    Article  Google Scholar 

  48. Filatrella, G., Pedersen, N.F., Lobb, C.J., Barbara, P.: Synchronization of underdamped Josephson-junction arrays. Eur. Phys. J. B 34(1), 3–8 (2003)

    Article  Google Scholar 

  49. Dhamala, M., Wiensefeld, K.: Generalized stability law for Josephson series arrays. Phys. Lett. A 292, 269–274 (2002)

    Article  MATH  Google Scholar 

  50. Pazó, D.: Thermodynamic limit of the first-order phase transition in the Kuramoto model. Phys. Rev. E 72, 046211 (2005)

    Article  MathSciNet  Google Scholar 

  51. Xie, F., Hu, G.: Clustering dynamics in globally coupled map lattices. Phys. Rev. E 56, 1567–1570 (1997)

    Article  Google Scholar 

  52. Williams, T.L.: Phase coupling by synaptic spread in chains of coupled neuronal oscillators. Science 258, 662–665 (1992)

    Article  Google Scholar 

  53. Wilson, M., Bower, J.M.: Cortical oscillations and temporal interactions in a computer simulation of piriform cortex. J. Neurophysiol. 67(4), 981–995 (1992)

    Google Scholar 

  54. Gray, C.M., Singer, W.: Stimulus-specific neuronal oscillations in orientation columns of cat visual cortex. Proc. Natl. Acad. Sci. U.S.A. 86, 1698–1702 (1989)

    Article  Google Scholar 

  55. Brown, B.H., Duthie, H.L., Horn, A.R., Smallwood, R.H.: A linked oscillator model of electrical activity of human small intestine. Am. J. Physiol. 229(2), 384–388 (1975)

    Google Scholar 

  56. Robertson-Dunn, B., Linkens, D.A.: A mathematical model of the slow wave electrical activity of the human small intestine. Med. Biol. Eng. 12(6), 750–758 (1974)

    Article  Google Scholar 

  57. Linkens, D.: Circuits and systems. IEEE Trans. 21(2), 294–300 (1974)

    Google Scholar 

  58. Brown, B.H., Nwong, K.K., Duthier, K.H.L., Whittaker, G.E., Franks, C.I.: Auto-correlation and visual analysis. Med. Biol. Eng. 9, 305–314 (1971)

    Article  Google Scholar 

  59. Duthie, H.L., Kwong, N.K., Brown, B.H., Whittaker, G.E.: Pacesetter potential of the human gastroduodenal junction. Gut 12, 250–256 (1971)

    Article  Google Scholar 

  60. Duthie, H.L., Brown, B.H., Robertson-Dunn, B., Kwong, N.K., Whittaker, G.E., Waterfall, W.: Electrical activity in the gastroduodenal area–slow waves in the proximal duodenum. A comparison of man and dog. Am. J. Dig. Dis. 17(4), 344–351 (1972)

    Article  Google Scholar 

  61. van der Pol, B.: On oscillation hysteresis in a triode generator with two degrees of freedom. Philos. Mag. 43(3), 700–719 (1922)

    Google Scholar 

  62. van der Pol, B.: The nonlinear theory of electric oscillations. Proc. Inst. Radio Eng. 22, 1051–1086 (1934)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fundamental Physics Laboratory, Group of Nonlinear Physics and Complex System, Department of Physics, Faculty of Science, University of Douala, Box 24 157, Douala, Cameroon

    R. Yamapi

  2. Institute for Development, Aging and Cancer, Tohoku University 4-1 Seiryocho, Aobaku, Sendai, 980-8575, Japan

    H. G. Enjieu Kadji

  3. Laboratorio Regionale SuperMat, CNR/INFM Salerno and Dipartimento di Scienze Biologiche ed Ambientali, Università del Sannio, via Port’Arsa 11, 82100, Benevento, Italy

    G. Filatrella

Authors
  1. R. Yamapi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H. G. Enjieu Kadji
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. G. Filatrella
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. Yamapi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yamapi, R., Enjieu Kadji, H.G. & Filatrella, G. Stability of the synchronization manifold in nearest neighbor nonidentical van der Pol-like oscillators. Nonlinear Dyn 61, 275–294 (2010). https://doi.org/10.1007/s11071-009-9648-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-009-9648-z

Keywords

Search

Navigation