Advertisement

Vibrational Resonance in Multistable and Excitable Systems

  • Shanmuganathan Rajasekar
  • Miguel A. F. Sanjuan
Chapter
Part of the Springer Series in Synergetics book series (SSSYN)

Abstract

As shown in Chap. 2 bistability is necessary for stochastic resonance. However, for vibrational resonance, bistability is not a necessary requirement. Even a monostable nonlinear system can exhibit double resonance. Thus, one wish to know the features of high-frequency induced resonance in double-well and triple-well systems. For this purpose, consider a multistable system (that is a system with more than one stable equilibrium point) driven by a single periodic force. In a multistable system, for sufficiently small values of f each of the coexisting orbits enclose only one equilibrium point. For higher values of f, a transition between the coexisting equilibrium states takes place. It is important to analyze the influence of such a transition on vibrational resonance. There is another class of systems called excitable systems . They have only one stable fixed point, but perturbations above a certain threshold induce large excursions in phase space, which take the form of spikes or pulses. That is, a rest condition can be transformed into a firing condition near the excitable threshold. Excitability is an essential characteristic of excitable media. Interestingly, many biological, physical and electronic circuit systems are excitable systems. How does vibrational resonance arise in excitable systems?

References

  1. 1.
    F.C. Moon, P.J. Holmes, J. Sound Vib. 65, 275 (1979)ADSCrossRefGoogle Scholar
  2. 2.
    N.M. Zimmerman, B. Golding, W.H. Haemmerle, Phys. Rev. Lett. 67, 1322 (1991)ADSCrossRefGoogle Scholar
  3. 3.
    E. Simiu, Chaotic Transitions in Deterministic and Stochastic Dynamical Systems (Princeton University Press, New Jersey, 2002)CrossRefzbMATHGoogle Scholar
  4. 4.
    I.I. Blekhman, P.S. Landa, Int. J. Nonlinear Mech. 39, 421 (2004)ADSCrossRefGoogle Scholar
  5. 5.
    S. Rajasekar, S. Jeyakumari, V. Chinnathambi, M.A.F. Sanjuan, J. Phys. A Math. Theor. 43, 465101 (2010)ADSCrossRefGoogle Scholar
  6. 6.
    S. Jeyakumari, V. Chinnathambi, S. Rajasekar, M.A.F. Sanjuan, Chaos 19, 043128 (2009)ADSCrossRefGoogle Scholar
  7. 7.
    B. Lindner, J. Garcia-Ojalvo, A. Neiman, L. Schimansky-Geier, Phys. Rep. 392, 321 (2004)ADSCrossRefGoogle Scholar
  8. 8.
    E. Ullner, A. Zaikin, J. Garcia-Ojalvo, R. Bascones, J. Kurths, Phys. Lett. A 312, 348 (2003)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    B. Deng, J. Wang, X. Wei, K.M. Tsang, W.L. Chan, Chaos 20, 013113 (2010)ADSCrossRefGoogle Scholar
  10. 10.
    R. FitzHugh, Biophys. J. 1, 445 (1961)ADSCrossRefGoogle Scholar
  11. 11.
    J. Nagumo, S. Arimoto, S. Yoshizawa, Proc. IRE 50, 2061 (1962)CrossRefGoogle Scholar
  12. 12.
    J. Keener, J. Snyder, Mathematical Physiology (Springer, New York, 1998)zbMATHGoogle Scholar
  13. 13.
    A.S. Mikhailov, Foundations of Synergetics (Springer, Berlin, 1994)CrossRefzbMATHGoogle Scholar
  14. 14.
    K. Park, Y.C. Lai, Z. Liu, A. Nachman, Phys. Lett. A 326, 391 (2004)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Y.C. Lai, Z. Liu, A. Nachman, L. Zhu, Int. J. Bifurcation Chaos 14, 3519 (2004)ADSCrossRefGoogle Scholar
  16. 16.
    X. Liang, L. Zhao, Z. Liu, Phys. Rev. E 84, 031916 (2011)ADSCrossRefGoogle Scholar
  17. 17.
    H. Zheng-You, Z. Yu-Rong, Chin. Phys. Lett. 28, 110505 (2011)CrossRefGoogle Scholar
  18. 18.
    V.N. Chizhevsky, G. Giacomelli, Phys. Rev. E 73, 022103 (2006)ADSCrossRefGoogle Scholar
  19. 19.
    V.N. Chizhevsky, G. Giacomelli, Phys. Rev. E 77, 051126 (2008)ADSCrossRefGoogle Scholar
  20. 20.
    A. Ichiki, Y. Tadokoro, M. Takanashi, J. Phys. A Math. Theor. 45, 385101 (2012)ADSCrossRefGoogle Scholar
  21. 21.
    C. Yao, Y. Liu, M. Zhan, Phys. Rev. E 83, 061122 (2011)ADSCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Shanmuganathan Rajasekar
    • 1
  • Miguel A. F. Sanjuan
    • 2
  1. 1.School of PhysicsBharathidasan UniversityTiruchirappalliIndia
  2. 2.Department of PhysicsUniversidad Rey Juan CarlosMóstoles, MadridSpain

Personalised recommendations