Stability and bifurcation in a neural network model with two delays

Wei, Junjie; Ruan, Shigui

doi:10.1016/s0167-2789(99)00009-3

Cited by 337 publications

(177 citation statements)

References 19 publications

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“…The dynamics of (19) are similar to the dynamics of (2), and ζ 1 = h(ζ 2 ) = 0 is the center manifold. The ζ 2 dynamics on this manifold are neutrally stable.…”

Section: Consensusmentioning

confidence: 84%

“…Certain nonlinear protocols to achieve consensus have been studied [1]. The bifurcation problem has been studied in neural networks; a Hopf-like bifurcation has been observed in a two cell autonomous system [20], and pitchfork and Hopf bifurcations have been studied in artificial neural networks [14], [19]. Some static bifurcations have been studied in load flow dynamics of power networks [9].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On Bifurcations in Nonlinear Consensus Networks

2011

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Abstract-The theory of consensus dynamics is widely employed to study various linear behaviors in networked control systems. Moreover, nonlinear phenomena have been observed in animal groups, power networks and in other networked systems. This inspires the development in this paper of two novel approaches to define distributed nonlinear dynamical interactions. The resulting dynamical systems are akin to higher-order nonlinear consensus systems. Over connected undirected graphs, the resulting dynamical systems exhibit various interesting behaviors that we rigorously characterize.

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“…The dynamics of (19) are similar to the dynamics of (2), and ζ 1 = h(ζ 2 ) = 0 is the center manifold. The ζ 2 dynamics on this manifold are neutrally stable.…”

Section: Consensusmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 99%

On Bifurcations in Nonlinear Consensus Networks

2011

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“…Therefore, it is often reasonable to incorporate different time delays into ODE systems to consider the maturation time and the intrinsic growth time of each species. In recent years, the effects of time delay due to maturation of the predator on the dynamics of predator-prey models have been studied by a number of authors (see, for example, [11,18,21,12,25] and references cited therein). But most of them investigated only the cases when the considered models have a single delay.…”

Section: Au(t)w(t)mentioning

confidence: 99%

“…It is well known that studies on dynamical systems not only involve a discussion of boundedness, stability and persistence, but also involve many dynamical behaviors such as periodic phenomenon, bifurcation and chaos. In particular, the properties of periodic solutions are of great interest arising through Hopf bifurcations in delayed systems, see Liu and Yuan [11], Wei and Ruan [21], Wei and Li [22] and Xiao and Li [23].…”

Section: +Bu(t) V (T) = V(t) −D + Eu(t−τ2)mentioning

confidence: 99%

Stability and Bifurcation Analysis for a Two-Competitor/One-Prey System With Two Delays

Guo-hu¹,

Yany²

2011

Journal of the Korean Mathematical Society

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Abstract. The present paper is concerned with a two-competitor/oneprey population system with Holling type-II functional response and two discrete delays. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the asymptotic stability of the positive equilibrium and existence of local Hopf bifurcations are investigated. Particularly, by applying the normal form theory and the center manifold reduction for functional differential equations (FDEs), explicit formulae determining the direction of bifurcations and the stability of bifurcating periodic solutions are derived. Finally, to verify our theoretical predictions, some numerical simulations are also included at the end of this paper.

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“…The problem of determining the distribution of roots to such polynomials is very complex and there are very few studies on this topic [11,38,39]. In this paper, we use an analytical approach proposed by Wei and Ruan [41] and Ruand and Wei [39].…”

Section: Introductionmentioning

confidence: 99%

Periodic oscillations in leukopoiesis models with two delays

Adimy

Crauste

Ruan

2006

Journal of Theoretical Biology

Self Cite

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The term leukopoiesis describes processes leading to the production and regulation of white blood cells. It is based on stem cells differentiation and may exhibit abnormalities resulting in severe diseases, such as cyclical neutropenia and leukemias. We consider a nonlinear system of two equations, describing the evolution of a stem cell population and the resulting white blood cell population. Two delays appear in this model to describe the cell cycle duration of the stem cell population and the time required to produce white blood cells. We establish sufficient conditions for the asymptotic stability of the unique nontrivial positive steady state of the model by analyzing roots of a second degree exponential polynomial characteristic equation with delay-dependent coefficients. We also prove the existence of a Hopf bifurcation which leads to periodic solutions. Numerical simulations of the model with parameter values reported in the literature demonstrate that periodic oscillations (with short and long periods) agree with observations of cyclical neutropenia in patients.

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Stability and bifurcation in a neural network model with two delays

Cited by 337 publications

References 19 publications

On Bifurcations in Nonlinear Consensus Networks

On Bifurcations in Nonlinear Consensus Networks

Stability and Bifurcation Analysis for a Two-Competitor/One-Prey System With Two Delays

Periodic oscillations in leukopoiesis models with two delays

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