Skip Navigation Links
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Global Synchronization of Large Ensembles of Pulse Oscillators with Time-Delay Coupling

Discontinuity, Nonlinearity, and Complexity 1(3) (2012) 253--261 | DOI:10.5890/DNC.2012.07.001

Vladimir V. Klinshov; Vladimir I. Nekorkin

Institute of Applied Physics, Nizhny Novgorod, Russia

Download Full Text PDF



In this paper we study the global synchronization in the ensembles of time-delay coupled pulse oscillators. We prove that the stability of the synchronization regime depends on the fulfillment of the simple inequality for the derivative of the phase reset curve. This finding is checked by numerical simulations which confirm the above inequality to be the sufficient condition for the synchronization. In the case of sin-shape phase reset curve it results in a stripped structure of the parameter space in which the areas with and without synchronic state alternate with the growth of the coupling delay.


The authors acknowledge the support from RFBR (grants No. 09-02-00719, 09-02-91061, 10-02-00643) and the Federal Target Program “Academic and teaching staff of innovative Russia” for 2009-2013 years (contracts No. P942, P1225, 02.740.11.5188, 14.740.11.0348).


  1. [1]  Schuster, H.G. and Wagner P. (1989), Mutual entrainment of two limit-cycle oscillators with time delayed coupling, Prog. Theor. Phys., 81, 939-945.
  2. [2]  Gerstner, W. (1996), Rapid Phase Locking in Systems of Pulse-Coupled Oscillators with Delays, Phys. Rev. Lett., 76(10), 1755-1-4.
  3. [3]  Yeung, M.K.S. and Strogatz, S.H. (1999), Time Delay in the Kuramoto Model of Coupled Oscillators, Phys. Rev. Lett., 82, 648-651.
  4. [4]  Choi, M.Y., Kim, H.J. and Kim, D. (2000), Synchronization in a system of globally coupled oscillators with time delay, Phys. Rev. E., 61, 371-381.
  5. [5]  Jiang Y. (2000), Globally coupled maps with time delay interactions, Phys. Lett. A., 267, 342-349.
  6. [6]  Earl, M.G. and Strogatz, S.H. (2003), Synchronization in oscillator networks with delayed coupling: A stability criterion,Phys. Rev. E., 67, 036204-1-4.
  7. [7]  Zillmer, R., Brunel, N. and Hansel, D. (2009), Very long transients, irregular firing, and chaotic dynamics in networks of randomly connected inhibitory integrate-and-fire neurons, Phys, Rev. E., 79, 031909-1-13.
  8. [8]  Wu, W. and Chen, T. (2009), Asymptotic Synchronization for Pulse-Coupled Oscillators with Delayed Excitatory Coupling Is Impossible, Advances in Computational Intell., AISC., 61, 45-51.
  9. [9]  Klinshov, V.V. and Nekorkin, V.I. (2011), Synchronization of time-delay coupled pulse oscillators,Chaos, Solitons & Fractals, 44(1-3), 98-107.
  10. [10]  Goel, P. and Ermentrot, B. (2002), Synchrony, stability, and firing patterns in pulse-coupled oscillators, Physica D, 163, 191-216.