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Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain


Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email:

Delay Effects on Amplitude Death, Oscillation Death, and Renewed Limit Cycle Behavior in Cyclically Coupled Oscillators

Journal of Applied Nonlinear Dynamics 10(3) (2021) 431--459 | DOI:10.5890/JAND.2021.09.006

Ryan Roopnarain, S. Roy Choudhury

Department of Mathematics, University of Central Florida

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The effects of a distributed `weak generic kernel' delay on cyclically coupled limit cycle and chaotic oscillators are considered. For coupled Van der Pol oscillators (and in fact, other oscillators as well) the delay can produce transitions from amplitude death(AD) or oscillation death (OD) to Hopf bifurcation-induced periodic behavior, with the delayed limit cycle shrinking or growing as the delay is varied towards or away from the bifurcation point respectively. The transition from AD to OD is mediated here via a pitchfork bifurcation, as seen earlier for other couplings as well. Also, the cyclically coupled undelayed van der Pol system here is already in a state of AD/OD, and introducing the delay allows both oscillations and AD/OD as the delay parameter is varied. This is in contrast to other limit cycle systems, where diffusive coupling alone does not result in the onset of AD/OD.\\ For systems where the individual oscillators are chaotic, such as a Sprott oscillator system or a coupled van der Pol-Rayleigh system with parametric forcing, the delay may produce AD/OD (as in the Sprott case), with the AD to OD transition now occurring via a transcritical bifurcation instead. However, this may not be possible, and the delay might just vary the attractor shape. In either of these situations however, increased delay strength tends to cause the system to have simpler behavior, streamlining the shape of the attractor, or shrinking it in cases with oscillations.


  1. [1]  Strogatz, S. (2012), Sync: How Order Emerges from Chaos in the Universe, Nature and Daily Life (Hyperion, New York, 2012).
  2. [2]  Boccaletti, S., Pisarchik, A.N., and del Genio, C.I. (2018), Synchronization: From Coupled Systems to Complex Networks (Cambridge U. Press, Cambridge, 2018).
  3. [3]  Pikovsky, A., Rosenblum, M.G., and Kurths, J. (2001), Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge U Press, Cambridge, 2001).
  4. [4]  Saxena, G., Prasad, A., and Ramaswamy, R. (2012), Amplitude death: the emergence of stationarity in coupled nonlinear systems, Phys. Rep., 521, 205.
  5. [5]  Koseska, A., Volkov, E., and Kurths, J. (2013), Oscillation quenching mechanisms: amplitude versus oscillation death, Phys. Rep., 531, 173.
  6. [6]  Ullner, E., Zaikin, A., Volkov, E., and Ojalvo, J.G. (2007), Multistability and clustering in a population of synthetic genetic oscillators via phase-repulsive cell-to-cell communication, Phys. Rev. Lett., 99, 148103.
  7. [7]  Suarez-Vargas, J.J., Gonzalez, J.A., and Stefanovska, A., and McClintock, P.V. (2009), Diverse routes to oscillation death in a coupled-oscillator system, Europhys. Lett., 85, 38008.
  8. [8]  Yoshimoto, M., Yoshikawa, K., and Mori, Y. (1993), Coupling among three chemical oscillators: synchronization, phase death, and frustration, Phys. Rev., E47, 864.
  9. [9]  Bar-Eli, K. (1985), On the stability of coupled chemical oscillators, Physica D, 14, 242.
  10. [10]  Aronson, D.G., Ermentrout, G.B., and Kopell, N. (1990), Amplitude response of coupled oscillators, Physica D, 41 403.
  11. [11]  Ramana Reddy, D.V., Sen, A., and Johnston, G.L. (1998), Time delay induced death in coupled limit cycle oscillators, Phys. Rev. Lett., 80, 5109.
  12. [12]  Konishi, K. (2003), Amplitude death induced by dynamic coupling, Phys. Rev. E, 68, 067202.
  13. [13]  Resmi, V., Ambika, G., and Amritkar, R.E. (2011), General mechanism for amplitude death in coupled systems, Phys. Rev. E, 84, 046212.
  14. [14]  Hens, C.R., Olusola, O.I., Pal, P., and Dana, S.K. (2013), Oscillation death in diffusively coupled oscillators by local repulsive link, Phs. Rev. E, 88, 034902.
  15. [15]  Koseska, A., Ullner, E., Volkov, E.I., Kurths, J., and Ojalvo, J.G. (2010), Cooperative differentiation through clustering in multicellular populations, J. Theoret. Bio., 263, 189.
  16. [16] Ruwisch, D., Bode, M., Volkov, D., and Volkov, E.I. (1999), Collective modes of three coupled relaxation oscillators: the influence of detuning, Int. J. Bifurc. Chaos Appl. Sci. Eng., 9, 1969.
  17. [17]  Rakshit, S., Bera, B., Majhi, S., Hens, C.R., and Ghosh, D. (2017), Basin stability measure of different steady states in coupled oscillators, Sci. Rep., 7, 45909.
  18. [18]  Banerjee, T., Biswas, D., Ghosh, D., Bandyopadhyay, B., and Kurths, J. (2018), Transition from homogeneous to inhomogeneous limit cycles: Effect of local filtering in coupled oscillators, Phys. Rev. E, 97, 042218.
  19. [19]  Turing, A. (1952), The chemical basis of morphogenesis, Philos. Trans. R. Soc. Lond. B, Biol. Sci., 237, 37.
  20. [20]  Koseska, A., Volkov, E., and Kurths, J. (2013), Transition from amplitude to oscillation death via Turing bifurcation, Phys. Rev. Lett., 111, 024103.
  21. [21]  Zou, W., Senthilkumar, D.V., Koseska, A., and Kurths, J. (2013), Generalizing the transition from amplitude to oscillation death in coupled oscillators, Phys. Rev. E, 88, 050901.
  22. [22]  Banerjee, T. and Biswas, D. (2013), Amplitude death and synchronized states in nonlinear time-delay systems coupled through mean-field diffusion, Chaos, 23, 043101.
  23. [23]  Banerjee, T. and Ghosh, D. (2014), Transition from amplitude to oscillation death under mean-field diffusive coupling Phys. Rev. E, 89, 052912.
  24. [24]  Banerjee, T. and Ghosh, D. (2014), Experimental observation of a transition from amplitude to oscillation death in coupled oscillators, Phys. Rev. E, 89, 062902.
  25. [25]  Hens, C.R., Pal, P., Bhowmick, S.K., Roy, P.K., Sen, A., and Dana, S.K. (2014), Diverse routes of transition from amplitude to oscillation death in coupled oscillators under additional repulsive links, Phys. Rev. E, 89, 032901.
  26. [26]  Nandan, M., Hens, C.R., Pal, P., and Dana, S.K. (2014), Transition from amplitude to oscillation death in a network of oscillators, Chaos, 24, 043103.
  27. [27]  Ermentrout, B. and Kopell, N. (1990), Oscillator death in systems of coupled neural oscillators, SIAM J. Appl. Math., 50, 125.
  28. [28]  Suzuki, N., Furusawa, C., and Kaneko, K. (2011), Oscillatory protein expression dynamics endows stem cells with robust differentiation potential, PLoS ONE, 6, e27232.
  29. [29]  Kamal, M.K.T., Sharma, P.R., and Shrimali, M.D. (2015), Suppression of oscillations in mean-field diffusion, Pramana, 84, 237.
  30. [30]  Sharma, P.R., Sharma, A., Shrimali, M.D., and Prasad, A. (2011), Targeting fixed-point solutions in nonlinear oscillators through linear augmentation, Phys. Rev. E, 83, 067201.
  31. [31]  Olusola, O.I., Njah, A.N., and Dana, S.K. (2013), Synchronization in chaotic oscillators by cyclic coupling, Eur. Phys. J. Spec. Top., 222, 927.
  32. [32]  Bera, B., Hens, C.R., Bhowmick, S., Pal, P., and Ghosh, D. (2016), Transition from homogeneous to inhomogeneous steady states in oscillators under cyclic coupling, Phys. Letters A, 380, 130.
  33. [33]  Roopnanrain, R. and Roy Choudhury, S. Distributed Delay Effects on Diffusively Coupled van der Pol Oscillators, and a Chaotic van der Pol-Rayleigh System, Discontinuity, Nonlinearity and Complexity, in press.
  34. [34]  Roy Choudhury S. and Roopnarain, R. Bifurcations and Amplitude Death from Distributed Delays in Coupled Landau-Stuart Oscillators and a Chaotic Parametrically Forced van der Pol-Rayleigh System, Intl. J. Bifs. and Chaos, submitted.
  35. [35]  Dudkowski, D., Jafari, S., Kapitaniak, T., Kuznetsov, N.V., Leonov, G.A., and Prasad, A. (2016), Hidden Attractors in Dynamical Systems, Phys. Reports, 637, 1.