zbMATH — the first resource for mathematics

Stability and bifurcation analysis in an amplitude equation with delayed feedback. (English) Zbl 1142.34377
Summary: An amplitude equation is considered. The linear stability of the equation with direct control is investigated, and hence a bifurcation set is provided in the appropriate parameter plane. It is found that there exist stability switches when delay varies, and the Hopf bifurcation occurs when delay passes through a sequence of critical values. Furthermore, the stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Finally, some numerical simulations are performed to illustrate the obtained results.

34K20 Stability theory of functional-differential equations
34K18 Bifurcation theory of functional-differential equations
Full Text: DOI
[1] Dhamala, M.; Jirsa, V.; Ding, M., Enhancement of neural synchrony by time delay, Phys rev lett, 92, 074104, (2004)
[2] Hassard, B.; Kazarinoff, N.; Wan, Y., Theory and applications of Hopf bifurcation, (1981), Cambridge University Press Cambridge · Zbl 0474.34002
[3] Ji, J.; Hansen, C., Stability and dynamics of a controlled van der Pol-Duffing oscillator, Chaos, solitons & fractals, 28, 555-570, (2006) · Zbl 1084.34040
[4] Jiang, W.; Wei, J., Bifurcation analysis in a limit cycle oscillator with delayed feedback, Chaos, solitons & fractals, 23, 817-831, (2005) · Zbl 1080.34054
[5] Jing, Z.; Yang, J.; Feng, W., Bifurcation and chaos in neural excitable system, Chaos, solitons & fractals, 27, 197-215, (2006) · Zbl 1102.37315
[6] Li, C.; Chen, G., Local stability and Hopf bifurcation in small-world delayed networks, Chaos, solitons & fractals, 20, 353-361, (2004) · Zbl 1045.34047
[7] Li, Y.; Zhou, X.; Wu, Y.; Zhou, M., Hopf bifurcation analysis of a tabu learning two-neuron model, Chaos, solitons & fractals, 29, 190-197, (2006) · Zbl 1110.68131
[8] Liao, X.; Ran, J., Hopf bifurcation in love dynamical models with nonlinear couples and time delays, Chaos, solitons & fractals, 31, 853-865, (2007) · Zbl 1152.34060
[9] Liu, C.; Tian, Y., Eliminating oscillations in the Internet by time-delayed feedback control, Chaos, solitons & fractals, 35, 878-887, (2008) · Zbl 1136.93426
[10] Meng, X.; Wei, J., Stability and bifurcation of mutual system with time delay, Chaos, solitons & fractals, 21, 729-740, (2004) · Zbl 1048.34122
[11] Niebur, E.; Schuster, H.; Kammen, D., Collective frequencies and meta stability in networks of limit-cycle oscillators with time delay, Phys rev lett, 67, 2753, (1991)
[12] Park, J.; Kwon, O., A novel criterion for delayed feedback control of time-delay chaotic systems, Chaos, solitons & fractals, 23, 495-501, (2005) · Zbl 1061.93507
[13] Pawlik, A.; Pikovsky, A., Control of oscillators coherence by multiple delayed feedback, Phys lett A, 358, 181-185, (2006) · Zbl 1142.93415
[14] Ramesh, M.; Narayanan, S., Chaos control of bonhoeffer – van der Pol oscillator using neural networks, Chaos, solitons, & fractals, 12, 2395-2405, (2001) · Zbl 1004.37067
[15] Rosenblum, M.; Pikovsky, A., Delayed feedback control of collective synchrony: an approach to suppression of pathological brain rhythms, Phys rev E, 70, 041904, (2004)
[16] Rosenblum, M.; Pikovsky, A., Controlling synchronization in an ensemble of globally coupled oscillators, Phys rev lett, 92, 114102, (2004)
[17] Ruan, S.; Wei, J., On the zeros of transcendental functions with applications to stability of delayed differential equations with two delays, Dyn contin discrete impuls syst ser A, math anal, 10, 863-874, (2003) · Zbl 1068.34072
[18] Song, Y.; Wei, J., Bifurcation analysis for chen’s system with delayed feedback and its application to control of chaos, Chaos, solitons & fractals, 22, 75-91, (2004) · Zbl 1112.37303
[19] Sun, J., Delay-dependent stability criteria for time-delay chaotic systems via time-delay feedback control, Chaos, solitons & fractals, 21, 143-150, (2004) · Zbl 1048.37509
[20] Wang, J.; Chen, L.; Fei, X., Analysis and control of the bifurcation of hodgkin – huxley model, Chaos, solitons & fractals, 31, 247-256, (2007) · Zbl 1140.37370
[21] Wang, Z.; Chu, T., Delay induced Hopf bifurcation in a simplified network congestion control model, Chaos, solitons & fractals, 28, 161-172, (2006) · Zbl 1105.34048
[22] Wei, J.; Zhang, C., Stability analysis in a first-order complex differential equations with delay, Nonlinear anal, 59, 657-671, (2004) · Zbl 1067.34082
[23] Yeung, M.; Strogatz, S., Time delay in the Kuramoto model of coupled oscillators, Phys rev lett, 82, 648-651, (1999)
[24] Zhang, Q.; Wei, X.; Xu, J., Stability of delayed cellular neural networks, Chaos, solitons & fractals, 31, 514-520, (2007) · Zbl 1156.34349
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.