Stability and Bifurcation Analysis of Synchronized Oscillations with Two Time Delays

Abstract

Abstract: The dynamics of synchronized oscillations with two time delays is discussed. The asymptotic stability of the positive equilibrium is investigated and Hopf bifurcations are demonstrated through linearizing the model at the positive equilibrium and analyzing the associated characteristic equation. Numerical simulations are carried out to support our theoretical analysis.

Key words: synchronized oscillations, time delays, stability, Hopf bifurcation, periodic solution

CLC Number:

O175.14

Li Chengxian, Liu Haihong. Stability and Bifurcation Analysis of Synchronized Oscillations with Two Time Delays[J]. JOURNAL OF QUJING NORMAL UNIVERSITY, 2016, 35(6): 14-19.

References

[1]Bernard S, Gonze D, Cajavec B, et al. Synchronization induced rhythmicity of circadian oscillators in the suprachiasmatic nucleus [J]. Plos Computational Biology, 2007 (3): 667-679.
[2]Brennan P A, Zufall F. Pheromonal communication in vertebrates [J]. Nature, 2006 (444): 308–315.
[3]Gonze D, Bernard S, Waltermann C, et al. Spontaneous synchronization of coupled circadian oscillators [J]. Biophysical Journal, 2005(89): 120-129.
[4]Vladimirov A G, Kozyreff G, Mandel P. Synchronization of weakly stable oscillators and semiconductor laser arrays [J]. Europhysics Letters, 2003(61): 613-619.
[5]Jiang Y J, Aerne B L,Smithers L, et al. Notch signalling and the synchronization of the somite segmentation clock [J]. Nature, 2000(408): 475-479.
[6]Elson R C, Selverston A I, Huerta R,et al. Synchronous behavior of two coupled biological neurons [J]. Physical Review Letters, 1998(81): 5692-5695.
[7]Mirollo R, Strogatz S. Synchronization of pulse-coupled biological oscillators [J]. Siam Journal on Applied Mathematics, 1990(50): 1645-1662.
[8]Glass L. Synchronization and rhythmic processes in physiology [J]. Nature, 2001(410): 277-284 .
[9]Young M W, Kay S. Time zones: a comparative genetics of circadian clocks [J]. Nature Reviews Genetics, 2001(2): 702-715.
[10]Chabot J R, Pedraza J, Luitel P, et al. Stochastic gene expression out-of-steady-state in the cyanobacterial circadian clock [J]. Nature, 2007(450), 1249-1252.
[11]Kerckhoffs R C, McCulloch A, Omens J, et al. Effects of biventricular pacing and scar size in a computational model of the failing heart with left bundle branch block [J]. Medical Image Analysis, 2009(13): 362-369.
[12]Roelfsema P R, Engel A K, Konig P, et al. Visuomotor integration is associated with zero time-lag synchronization among cortical areas [J]. Nature, 1997(385): 157-161.
[13]Schoffelen J M, Oostenveld R, Fries P. Neuronal coherence as a mechanism of effective corticospinal interaction [J]. Science, 2005(308): 111-113.
[14]Womelsdorf T, Schoffelen J M,Oostenveld R, et al. Modulation of neuronal interactions through neuronal synchronization [J]. Science, 2007(316): 1609–1612.
[15]Kim J, Shin D, Jung S, et al. A design principle underlying the synchronization of oscillations in cellular systems [J]. Journal of Cell Science, 2010(123): 537–543.
[16]Roxin A, Brunel N, Hansel D. Role of delays in shaping spatiotemporal dynamics of neuronal activity in large networks [J]. Physical Review Letters, 2005(94):1-4.
[17]Mao X C. Stability and Hopf bifurcation analysis of a pair of three-neuron loops with time delays [J]. Nonlinear Dynamics, 2012(68): 151-159.
[18] Ruan S, Wei J. On the zeros of transcendental functions with applications to stability of delayed differential equations with two delays [J]. Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 2003(10) : 863–874.
[19]Hu H Y, Wang Z H. Dynamics of Controlled Mechanical Systems with Delayed Feedback [M]. New York: Springer-Verlag, 2002: 1-114.
[20]Hale J K, Lungel S M V. Introduction to Functional Differential Equations [M]. New York : Springer-Verlag ,1993:11-66.