Stability and Hopf bifurcation analysis on a four-neuron BAM neural network with distributed delays

In this paper, a four-neuron BAM neural network with distributed delays is considered, where kernels are chosen as weak kernels. Its dynamics is studied in terms of local stability analysis and Hopf bifurcation analysis. By choosing the average delay as a bifurcation parameter and analyzing the associated characteristic equation, Hopf bifurcation occurs when the bifurcation parameter passes through some exceptive values. The stability of bifurcating periodic solutions and a formula for determining the direction of Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Finally, numerical simulation results are given to validate the theorem obtained.     

Synchronized Hopf bifurcation analysis in a neural network model with delays

We consider the synchronized periodic oscillation in a ring neural network model with two different delays in self-connection and nearest neighbor coupling. Employing the center manifold theorem and normal form method introduced by Hassard et al., we give an algorithm for determining the Hopf bifurcation properties. Using the global Hopf bifurcation theorem for FDE due to Wu and Bendixson's criterion for high-dimensional ODE due to Li and Muldowney, we obtain several groups of conditions that guarantee the model have multiple synchronized periodic solutions when the transfer coefficient or time delay is sufficiently large.     

Hopf bifurcation and stability for a neural network model with mixed delays

A generalized model of the two-neuron network with mixed delays is studied. The main purpose of this paper is to explore the linear stability of the trivial solution and Hopf bifurcation of a two-neuron network with continuous and discrete delays. The general formula of the direction, the estimation formula of period and stability of bifurcated periodic solutions are also studied. Finally, the numerical simulations are given to illustrate the theoretical analysis.     

Hopf bifurcation in Cohen–Grossberg neural network with distributed delays

In this paper, we discuss the stability and bifurcation of the distributed delays Cohen–Grossberg neural networks with two neurons. By choosing the average delay as a bifurcation parameter, we prove that Hopf bifurcation occurs. The stability of bifurcating periodic solutions and the direction of Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Finally, numerical simulation results are given to support the theoretical predictions.     

Global existence of periodic solutions on a simplified BAM neural network model with delays

A simplified n-dimensional BAM neural network model with delays is considered. Some results of Hopf bifurcations occurring at the zero equilibrium as the delay increases are exhibited. Global existence of periodic solutions are established using a global Hopf bifurcation result of Wu [Wu J. Symmetric functional-differential equations and neural networks with memory. Trans Am Math Soc 1998;350:4799–838], and a Bendixson criterion for higher dimensional ordinary differential equations due to Li and Muldowney [Li MY, Muldowney J. On Bendixson’s criterion. J Differ Equations 1994;106:27–39]. Finally, computer simulations are performed to illustrate the analytical results found.     

Global exponential stability analysis on impulsive BAM neural networks with distributed delays

Using M-matrix and topological degree tool, sufficient conditions are obtained for the existence, uniqueness and global exponential stability of the equilibrium point of bidirectional associative memory (BAM) neural networks with distributed delays and subjected to impulsive state displacements at fixed instants of time by constructing a suitable Lyapunov functional. The results remove the usual assumptions that the boundedness, monotonicity, and differentiability of the activation functions. It is shown that in some cases, the stability criteria can be easily checked. Finally, an illustrative example is given to show the effectiveness of the presented criteria.     

Hopf bifurcation and stability analysis on discrete-time Hopfield neural network with delay

In this paper, a discrete-time Hopfield neural network with delay is considered. We give some sufficient conditions ensuring the local stability of the equilibrium point for this model. By choosing the delay as a bifurcation parameter, we demonstrated that Neimark–Sacker bifurcation (or Hopf bifurcation for map) would occur when the delay exceeds a critical value. A formula for determining the direction bifurcation and stability of bifurcation periodic solutions is given by applying the normal form theory and the center manifold theorem. Some numerical simulations for justifying the theoretical results are also provided.     

Global Hopf bifurcation in the Leslie-Gower predator-prey model with two delays

In this paper, the Leslie-Gower predator-prey system with two delays is investigated. By choosing the delay as a bifurcation parameter, we show that Hopf bifurcations can occur as the delay crosses some critical values. In addition, special attention is paid to the global continuation of local Hopf bifurcations. Using a global Hopf bifurcation theorem for functional differential equations, we show the global existence of periodic solutions.     

Hopf bifurcation in bidirectional associative memory neural networks with delays: analysis and computation

In addition to giving some delay-independent conditions for global stability of the bidirectional associative memory neural networks with delays, we perform analysis of local stability and Hopf bifurcation. We also work out an algorithm for determining the direction and stability of the bifurcated periodic solutions.     

Hopf bifurcation analysis of two neurons with three delays

Using the system parameter instead of the delays as the bifurcation parameter, linear stability and Hopf bifurcation including its direction and stability of a two-neuron network with three delays are investigated in this paper. The main tools to obtain our results are the normal form method and the center manifold theory introduced by Hassard. Simulations show that the theoretically predicted values are in excellent agreement with the numerically observed behavior.     

Hopf bifurcation analysis for congestion control with heterogeneous delays

In this paper, a congestion control algorithm with heterogeneous delays is considered. Local stability of the equilibrium solution of this algorithm is investigated based on analyzing the corresponding transcendental characteristic equation. Especially, using one of the system parameters of congestion control algorithm as the bifurcation parameter, when the system parameter exceeds a critical value, the congestion control algorithm undergoes a supercritical Hopf bifurcation, and the explicit formulae determining the stability and the direction of periodic solutions bifurcating from the equilibrium are obtained by applying Hassard et al.’s approaches. Finally, some numerical simulations are performed to verify the theoretical results.     

Hopf bifurcation in a three-species system with delays

A kind of three-species system with Holling II functional response and two delays is introduced. Its local stability and the existence of Hopf bifurcation are demonstrated by analyzing the associated characteristic equation. By using the normal form method and center manifold theorem, explicit formulas to determine the direction of the Hopf bifurcation and the stability of bifurcating periodic solution are also obtained. In addition, the global existence results of periodic solutions bifurcating from Hopf bifurcations are established by using a global Hopf bifurcation result. Numerical simulation results are also given to support our theoretical predictions.     

Multiple Hopf bifurcations of symmetric BAM neural network model with delay

A symmetric BAM neural network model with delay is considered. Some results of Hopf bifurcations occurring at the zero equilibrium as the delay increases are exhibited. The existence of multiple periodic solutions is established using a symmetric Hopf bifurcation result of Wu [J. Wu, Symmetric functional differential equations and neural networks with memory, Transactions of the American Mathematical Society 350 (12) (1998) 4799–4838].     

一类具有Leakage时滞的惯性Cohen-Grossberg神经网络的全局指数稳定性和Hopf分支

研究一类具有Leakage时滞的惯性Cohen-Grossberg神经网络模型.通过构造适当的Lyapunov泛函得到了平衡点全局指数稳定的充分条件.通过分析特征方程,讨论了系统平衡点的局部稳定性,得出了系统Hopf分支存在的充分条件.最后对所得理论结果进行了数值模拟.     

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

We consider a four-neuron ring with self-feedback and delays. By analyzing the associated characteristic equation, linear stability is investigated and Hopf bifurcations are demonstrated, as well as the stability and direction of the Hopf bifurcation are determined by employing the normal form method and the center manifold reduction. Numerical simulations are presented to illustrate the results.     

Stability and Hopf bifurcation analysis on Goodwin model with three delays

In this paper, a class of Goodwin models with three delays is dealt. The dynamic properties including stability and Hopf bifurcations are studied. Firstly, we prove analytically that the addressed system possesses a unique positive equilibrium point. Moreover, using the Cardano’s formula for the third degree algebra equation, the distribution of characteristic roots is proposed. And then, the sum of the delays is chosen as the bifurcation parameter and it is demonstrated that the Hopf bifurcation would occur when the delay exceeds a critical value. Finally, a numerical simulation for justifying the theoretical results is also provided.     

Global exponential stability of fuzzy BAM neural networks with transmission delays

In this paper, a class of fuzzy bidirectional associated memory (BAM) neural networks with transmission delays are studied. Some sufficient conditions are established for the existence , uniqueness and global exponential stability of equilibrium point. The sufficient conditions are easy to verify at pattern recognition and automatic control. Finally, an example is given to show feasibility and effectiveness of our results.     

Stability and bifurcation analysis for a neural network model with discrete and distributed delays

In this paper, a two‐neuron network with both discrete and distributed delays is considered. With the corresponding characteristic equation analyzed, the local stability of the trivial equilibrium is investigated. With the discrete time delay taken as a bifurcation parameter, the existence of Hopf bifurcation is established. Moreover, formulae for determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are derived. Finally, numerical simulations are carried out to illustrate the main results and further to exhibit that there is a characteristic sequence of bifurcations leading to a chaotic dynamics, which implies that the system admits rich and complex dynamics. Copyright © 2012 John Wiley & Sons, Ltd.     

Hopf bifurcation analysis in a synaptically coupled FHN neuron model with delays

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

Stability and bifurcation of a Cohen-Grossberg neural network with discrete delays

A Cohen-Grossberg neural network with discrete delays is investigated in this paper. The qualitative analysis is given for the system and it is found that the system undergoes a sequence of Hopf bifurcations by choosing the discrete time delay as a bifurcation parameter. Moreover, by applying the normal form theory and the center manifold theorem, the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions are obtained. Numerical simulations are given to illustrate the obtained results.