Stability and Hopf bifurcation analysis on a ring of four neurons with delays

Hu, Haijun; Huang, Lihong

doi:10.1016/j.amc.2009.03.052

Cited by 41 publications

(22 citation statements)

References 24 publications

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“…Motivated by Lemma 2.3 of Ruan and Wei [15], Lemma 2.4 of Li and Wei [16], and Lemma 2.5 of Hu and Huang [17], we obtain the following conclusions.…”

Section: Lemma 21 Consider the Exponential Polynomialmentioning

confidence: 80%

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Hopf bifurcation analysis in synaptically coupled HR neurons with two time delays

2010

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This paper presents an investigation of stability and Hopf bifurcation of a synaptically coupled nonidentical HR model with two time delays. By regarding the half of the sum of two delays as a parameter, we first consider the existence of local Hopf bifurcations, and then derive explicit formulas for determining the direction of the Hopf bifurcations and the stability of bifurcating periodic solutions, using the normal form method and center manifold theory. Finally, numerical simulations are carried out for supporting theoretical analysis results.

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“…Motivated by Lemma 2.3 of Ruan and Wei [15], Lemma 2.4 of Li and Wei [16], and Lemma 2.5 of Hu and Huang [17], we obtain the following conclusions.…”

Section: Lemma 21 Consider the Exponential Polynomialmentioning

confidence: 80%

“…17) then ±iω + k are a pair of purely imaginary roots of the first equation of (2.6) with τ = τ (j ) k (+) (k = 1, 2, 3; j = 0, 1, 2, . .…”

mentioning

confidence: 99%

Hopf bifurcation analysis in synaptically coupled HR neurons with two time delays

2010

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“…The properties of the periodic solutions appearing through the Hopf bifurcation in delay differential equations are of great interest, see [1,2,4,24,23,29,15]. Further investigations such as the stability, direction and global existence of the periodic solutions bifurcating from the local Hopf bifurcations are essential in order to fully understand the periodic phenomenon of system (1.3).…”

Section: Discussionmentioning

confidence: 98%

Stability and Hopf bifurcation in a delayed predator–prey system with stage structure for prey

Huang

2010

Nonlinear Analysis: Real World Applications

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“…Motivated by Lemma 2.3 of the work of Ruan and Wei [53], Lemma 2.4 of the work of Li and Wei [52], Lemma 2.5 of the paper of Hu and Huang [54], and Theorem 2.1 of Fan and Hong [36], we obtain following conclusions.…”

Section: ((R-h) Hypothesis) (I) If One Of the Followings Holds: (A) Smentioning

confidence: 99%

Bifurcation structure of two coupled FHN neurons with delay

Tehrani

Razvan

2015

Mathematical Biosciences

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This paper presents an investigation of the dynamics of two coupled non-identical FitzHugh-Nagumo neurons with delayed synaptic connection. We consider coupling strength and time delay as bifurcation parameters, and try to classify all possible dynamics which is fairly rich. The neural system exhibits a unique rest point or three ones for the different values of coupling strength by employing the pitchfork bifurcation of non-trivial rest point. The asymptotic stability and possible Hopf bifurcations of the trivial rest point are studied by analyzing the corresponding characteristic equation. Homoclinic, fold, and pitchfork bifurcations of limit cycles are found. The delay-dependent stability regions are illustrated in the parameter plane, through which the double-Hopf bifurcation points can be obtained from the intersection points of two branches of Hopf bifurcation. The dynamical behavior of the system may exhibit one, two, or three different periodic solutions due to pitchfork cycle and torus bifurcations (Neimark-Sacker bifurcation in the Poincare map of a limit cycle), of which detection was impossible without exact and systematic dynamical study. In addition, Hopf, double-Hopf, and torus bifurcations of the non trivial rest points are found. Bifurcation diagrams are obtained numerically or analytically from the mathematical model and the parameter regions of different behaviors are clarified.

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Stability and Hopf bifurcation analysis on a ring of four neurons with delays

Cited by 41 publications

References 24 publications

Hopf bifurcation analysis in synaptically coupled HR neurons with two time delays

Hopf bifurcation analysis in synaptically coupled HR neurons with two time delays

Stability and Hopf bifurcation in a delayed predator–prey system with stage structure for prey

Bifurcation structure of two coupled FHN neurons with delay

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