HOPF BIFURCATION OF A TWO-NEURON NETWORK WITH DIFFERENT DISCRETE TIME DELAYS

2005 ◽  
Vol 15 (05) ◽  
pp. 1589-1601 ◽  
Author(s):  
SHAOWEN LI ◽  
XIAOFENG LIAO ◽  
CHUNGUANG LI ◽  
KWOK-WO WONG
Keyword(s):  
Hopf Bifurcation ◽  
Asymptotic Stability ◽  
Periodic Solutions ◽  
Discrete Time ◽  
Time Delays ◽  
Bifurcation Parameter ◽  
Neuron Network ◽  
Bifurcation Theorem ◽  
Nyquist Criterion ◽  
Delay Dependent

In this paper, a two-neuron network with different time delays is investigated. By analyzing the associated characteristic equation, we obtain the conditions for delay-dependent and delay-independent asymptotic stability, respectively. Furthermore, we find that if the delay is used as a bifurcation parameter, Hopf bifurcation would occur. The direction and stability of the bifurcating periodic solutions are determined by using the Nyquist criterion and the graphical Hopf bifurcation theorem. Some examples are included to illustrate our results.

Graphical Hopf Bifurcation of a Filippov HTLV-1 Model With Delay in Cytotoxic T Cells Response

2018 ◽  
Vol 140 (9) ◽  
Author(s):  
Elham Shamsara ◽  
Zahra Afsharnezhad ◽  
Elham Javidmanesh
Keyword(s):  
T Cells ◽  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Bifurcation Theory ◽  
Cytotoxic T Cells ◽  
Dynamical Behavior ◽  
Bifurcation Theorem ◽  
Nyquist Criterion ◽  
The Stability ◽  
Frequency Domain Approach

In this paper, we present a discontinuous cytotoxic T cells (CTLs) response for HTLV-1. Moreover, a delay parameter for the activation of CTLs is considered. In fact, a system of differential equation with discontinuous right-hand side with delay is defined for HTLV-1. For analyzing the dynamical behavior of the system, graphical Hopf bifurcation is used. In general, Hopf bifurcation theory will help to obtain the periodic solutions of a system as parameter varies. Therefore, by applying the frequency domain approach and analyzing the associated characteristic equation, the existence of Hopf bifurcation by using delay immune response as a bifurcation parameter is determined. The stability of Hopf bifurcation periodic solutions is obtained by the Nyquist criterion and the graphical Hopf bifurcation theorem. At the end, numerical simulations demonstrated our results for the system of HTLV-1.

scholarly journals Stability and Andronov-Hopf Bifurcation of a System with Three Time Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Svetoslav Nikolov
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Discrete Time ◽  
Local Stability ◽  
Time Delays ◽  
General System ◽  
System Modelling ◽  
Mirna Regulation ◽  
Bifurcation Theorem ◽  
Limit Cycle Bifurcation

A general system of three autonomous ordinary differential equations with three discrete time delays is considered. With respect to the delays, we investigate the local stability of equilibria by analyzing the corresponding characteristic equation. Using the Hopf bifurcation theorem, we predict the occurrence of a limit cycle bifurcation for the time delay parameters. Thus, some new mathematical results are obtained. Finally, the above mentioned criteria are applied to a system modelling miRNA regulation.

scholarly journals Hopf Bifurcation of a Mathematical Model for Growth of Tumors with an Action of Inhibitor and Two Time Delays

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
Bao Shi ◽  
Fangwei Zhang ◽  
Shihe Xu
Keyword(s):  
Mathematical Model ◽  
Hopf Bifurcation ◽  
Time Delays ◽  
Cell Loss ◽  
Bifurcation Parameter ◽  
Discrete Delays

A mathematical model for growth of tumors with two discrete delays is studied. The delays, respectively, represent the time taken for cells to undergo mitosis and the time taken for the cell to modify the rate of cell loss due to apoptosis and kill of cells by the inhibitor. We show the influence of time delays on the Hopf bifurcation when one of delays is used as a bifurcation parameter.

STABILITY AND HOPF BIFURCATION ON A TWO-NEURON SYSTEM WITH TIME DELAY IN THE FREQUENCY DOMAIN

2007 ◽  
Vol 17 (04) ◽  
pp. 1355-1366 ◽  
Author(s):  
WENWU YU ◽  
JINDE CAO
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Frequency Domain ◽  
Harmonic Balance ◽  
Neuron System ◽  
Bifurcation Theorem ◽  
Nyquist Criterion ◽  
The Stability ◽  
Frequency Domain Approach ◽  
System With Time Delay

In this paper, a general two-neuron model with time delay is considered, where the time delay is regarded as a parameter. It is found that Hopf bifurcation occurs when this delay passes through a sequence of critical value. By analyzing the characteristic equation and using the frequency domain approach, the existence of Hopf bifurcation is determined. The stability of bifurcating periodic solutions are determined by the harmonic balance approach, Nyquist criterion and the graphic Hopf bifurcation theorem. Numerical results are given to justify the theoretical analysis.

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