Nonlinear Dynamics

, Volume 73, Issue 3, pp 1253–1269 | Cite as

Synchronization and chaos control by quorum sensing mechanism

  • Liuxiao GuoEmail author
  • Manfeng Hu
  • Zhenyuan Xu
  • Aihua Hu
Original Paper


Diverse rhythms are generated by thousands of oscillators that somehow manage to operate synchronously. By using mathematical and computational modeling, we consider the synchronization and chaos control among chaotic oscillators coupled indirectly but through a quorum sensing mechanism. Some sufficient criteria for synchronization under quorum sensing are given based on traditional Lyapunov function method. The Melnikov function method is used to theoretically explain how to suppress chaotic Lorenz systems to different types of periodic oscillators in quorum sensing mechanics. Numerical studies for classical Lorenz and Rössler systems illustrate the theoretical results.


Synchronization Quorum sensing Chaotic oscillators Collective behavior 



The paper is supported by the National Science Foundation of PR China under Grants No 11002061, 11202084, 10901073 and “the Fundamental Research Funds for the Central Universities”.


  1. 1.
    Perora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824 (1990) MathSciNetCrossRefGoogle Scholar
  2. 2.
    Guo, L., Xu, Z.: Hölder continuity of two types of generalized synchronization manifold. Chaos 18, 033134 (2008) MathSciNetCrossRefGoogle Scholar
  3. 3.
    Mahmoud, G.M., Mahmoud, E.E.: Complete synchronization of chaotic complex nonlinear systems with uncertain parameters. Nonlinear Dyn. 62, 875–882 (2010) zbMATHCrossRefGoogle Scholar
  4. 4.
    Zhou, J., Wu, Q., Xiang, L.: Impulsive pinning complex dynamical networks and applications to firing neuronal synchronization. Nonlinear Dyn. 69, 1393–1403 (2012) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Odibat, Z.M.: Adaptive feedback control and synchronization of non-identical chaotic fractional order systems. Nonlinear Dyn. 60, 479–487 (2012) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Guo, L., Xu, Z., Hu, M.: Adaptive projective synchronization with different scaling factors in networks. Chin. Phys. B 17, 4067–4072 (2008) CrossRefGoogle Scholar
  7. 7.
    Huygens, C.: Oeuvres complètes de Christiaan Huygens vol. 17. Nijhoff, The Hague (1932) Google Scholar
  8. 8.
    Pikovsky, A., Rosenblum, M., Kurths, J.: Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge, Cambridge University Press (2001) CrossRefGoogle Scholar
  9. 9.
    Vladimirov, A.G., Kozyreff, G., Mandel, P.: Synchronization of weakly stable oscillators and semiconductor laser arrays. Europhys. Lett. 61, 613–619 (2003) CrossRefGoogle Scholar
  10. 10.
    Wiesenfeld, K., Colet, P., Strogatz, S.: Synchronization transitions in a disordered Josephson series array. Phys. Rev. Lett. 76, 404–407 (1996) CrossRefGoogle Scholar
  11. 11.
    Boccaletti, S., Grebogi, C., Lai, Y.C., Mancini, H., Maza, D.: The control of chaos: theory and applications. Phys. Rep. 329, 103–197 (2000) MathSciNetCrossRefGoogle Scholar
  12. 12.
    Chen, G., Dong, X.: From Chaos to Order: Perspectives Methodologies and Applications. World Scientific, Singapore (1998) zbMATHGoogle Scholar
  13. 13.
    Lu, J., Wu, X., Lü, J.: Synchronization of a unified chaotic system and the application in secure communication. Phys. Lett. A 305, 365–370 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Pikovsky, A.S., Rosenblum, M.G., Osipov, G.V., Kurths, J.: Phase synchronization of chaotic oscillators by external driving. Physica D 104, 219–238 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Pikovsky, A.S., Zaks, M., Rosenblum, M., Osipov, G., Kurths, J.: Phase synchronization of chaotic oscillations in terms of periodic orbits. Chaos 7, 680–688 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Zhou, C.S., Kurths, J.: Noise-induced phase synchronization and synchronization transitions in chaotic oscillators. Phys. Rev. Lett. 88, 230602 (2002) CrossRefGoogle Scholar
  17. 17.
    Katriel, G.: Synchronization of oscillators coupled through an environment. Physica D 237, 2933–2944 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Resmi, V., Ambika, G., Amritkar, R.E.: Synchronized states in chaotic systems coupled indirectly through dynamic environment (2010). arXiv:0910.2382v2 [nlin.CD]
  19. 19.
    Garcia-Ojalvo, J., Elowitz, M.B., Strogatz, S.H.: Modeling a synthetic multicellular clock: repressilators coupled by quorum sensing. Proc. Natl. Acad. Sci. USA 101, 10955–10960 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Danino, T., Mondragón-Palomino, O., Tsimring, L., Hasty, J.: A synchronized quorum of genetic clocks. Nature 463, 326–330 (2010) CrossRefGoogle Scholar
  21. 21.
    Misra, J.C., Mitra, A.: Synchronization among tumor-like cell aggregations coupled by quorum sensing: a theoretical study. Comput. Math. Appl. 55, 1842–1853 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    McMillen, D., Kopell, N., Hasty, J., Collins, J.J.: Synchronizing genetic relaxation oscillators by intercell signaling. Proc. Natl. Acad. Sci. USA 99, 679–684 (2002) CrossRefGoogle Scholar
  23. 23.
    Ott, E., Grebogi, C., Yorke, J.A.: Controlling chaos. Phys. Rev. Lett. 64, 1196–1199 (1990) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Leloup, J.C., Goldbeter, A.: A model for circadian rhythms in drosophila incorporating the formation of a complex between the PER and TIM proteins. J. Biol. Rhythms 13, 70–87 (1998) CrossRefGoogle Scholar
  25. 25.
    Glossop, N.R., Lyons, L.C., Hardin, P.E.: Interlocked feedback loops within the drosophila circadian oscillator. Science 286, 766–768 (1999) CrossRefGoogle Scholar
  26. 26.
    Chialvo, D.R., Gilmour, J., Jalife, J.: Low dimensional chaos in cardiac tissue. Nature 343, 653–657 (1990) CrossRefGoogle Scholar
  27. 27.
    Courtemanche, M., Glass, L., Rosengarten, M.D., Goldberger, A.L.: Beyond pure parasystole: promises and problems in modeling complex arrhythmias. Am. J. Physiol. 257, H693–H706 (1989) Google Scholar
  28. 28.
    Chen, A.: Modeling a synthetic biological chaotic system: relaxation oscillators coupled by quorum sensing. Nonlinear Dyn. 63, 711–718 (2011) CrossRefGoogle Scholar
  29. 29.
    Guo, L., Zhenyuan, X.: Adaptive coupled synchronization of non-autonomous systems in ring networks. Chin. Phys. B. 17, 836–841 (2008) zbMATHCrossRefGoogle Scholar
  30. 30.
    Li, J., Zhao, X., Liu, Z.: Generalized Hamilton Systems Theory and Its Applications. Science, Beijing (1997) Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Liuxiao Guo
    • 1
    Email author
  • Manfeng Hu
    • 1
  • Zhenyuan Xu
    • 1
  • Aihua Hu
    • 1
  1. 1.School of ScienceJiangnan UniversityWuxiChina

Personalised recommendations