Global Hopf bifurcation analysis on a BAM neural network with delays

A BAM neural network with three neurons is considered. Sufficient conditions for the system to have multiple periodic solutions are obtained when the sum of delays is sufficiently large. Numerical simulations are presented to support the theoretical results found.     

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.     

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.     

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.     

Stability in impulsive Cohen-Grossberg-type BAM neural networks with distributed delays

In this paper, a class of impulsive Cohen-Grossberg-type bi-directional associative memory (BAM) neural networks with distributed delays is investigated. By establishing an integro-differential inequality with impulsive initial conditions and employing the homeomorphism theory, the M-matrix theory and inequality technique, some new general sufficient conditions ensuring the existence, uniqueness and global exponential stability of equilibrium point for impulsive Cohen-Grossberg-type BAM neural networks with distributed delays are obtained. In particular, the estimate of the exponential convergence rate is also provided, which depends on the system parameters and impulsive disturbed intension. An example is given to show the effectiveness of the results obtained here.     

Stability and bifurcation of a two-neuron network with distributed time delays

In this paper we study the stability and bifurcation of the trivial solution of a two-neuron network model with distributed time delays. This model consists of two identical neurons, each possessing nonlinear instantaneous self-feedback and connected to the other neuron with continuously distributed time delays. We first examine the local asymptotic stability of the trivial solution by studying the roots of the corresponding characteristic equation, and then describe the stability and instability regions in the parameter space consisting of the self-feedback strength and the product of the connection strengths between the neurons. It is further shown that the trivial solution may lose its stability via a certain type of bifurcation such as a Hopf bifurcation or a pitchfork bifurcation. In addition, the criticality of Hopf bifurcation is investigated by means of the normal form theory. We also provide numerical evidence to support our theoretical analyses.     

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.     

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 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.     

Stability and Hopf bifurcation for a regulated logistic growth model with discrete and distributed delays

In this paper, we investigate the stability and Hopf bifurcation of a new regulated logistic growth with discrete and distributed delays. By choosing the discrete delay τ as a bifurcation parameter, we prove that the system is locally asymptotically stable in a range of the delay and Hopf bifurcation occurs as τ crosses a critical value. Furthermore, explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived by normal form theorem and center manifold argument. Finally, an illustrative example is also given to support the theoretical results.     

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.     

Stability and Hopf bifurcation of a delayed reaction–diffusion neural network

In this paper, a delayed reaction–diffusion neural network with Neumann boundary conditions is investigated. By analyzing the corresponding characteristic equations, the local stability of the trivial uniform steady state is discussed. The existence of Hopf bifurcation at the trivial steady state is established. Using the normal form theory and the center manifold reduction of partial function differential equations, explicit formulae are derived to determine the direction and stability of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the main results. Copyright © 2011 John Wiley & Sons, Ltd.     

Stability and Hopf bifurcation analysis of a prey–predator system with two delays

In this paper, we have considered a prey–predator model with Beddington-DeAngelis functional response and selective harvesting of predator species. Two delays appear in this model to describe the time that juveniles take to mature. Its dynamics are studied in terms of local analysis and Hopf bifurcation analysis. By analyzing the associated characteristic equation, its linear stability is investigated and Hopf bifurcations are demonstrated. The stability and direction of the Hopf bifurcation are determined by applying the normal form method and the center manifold theory. Numerical simulation results are given to support the theoretical predictions.     

Stability and Hopf bifurcation of a delayed Cohen–Grossberg neural network with diffusion

In this paper, a delayed Cohen–Grossberg neural network with diffusion under homogeneous Neumann boundary conditions is investigated. By analyzing the corresponding characteristic equation, the local stability of the trivial uniform steady state and the existence of Hopf bifurcation at the trivial steady state are established, respectively. By using the normal form theory and the center manifold reduction of partial function differential equations, formulae are derived to determine the direction of bifurcations and the stability of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the main results. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

Stability and Hopf bifurcation analysis in bidirectional ring network model

Using the system parameter instead of the delay as the bifurcation parameter, linear stability and Hopf bifurcation analysis including its direction and stability of bidirectional ring network model are investigated in this paper. The main tools to obtain our results are the normal form method and center manifold theory. Numerical simulations show that the theoretically predicted values are in excellent agreement with the numerically observed behavior.     

Stability analysis of Cohen–Grossberg neural network with both time-varying and continuously distributed delays

In this paper, the Cohen–Grossberg neural network model with both time-varying and continuously distributed delays is considered. Without assuming both global Lipschitz conditions on these activation functions and the differentiability on these time-varying delays, applying the idea of vector Lyapunov function, M-matrix theory and inequality technique, several new sufficient conditions are obtained to ensure the existence, uniqueness, and global exponential stability of equilibrium point for Cohen–Grossberg neural network with both time-varying and continuously distributed delays. These results generalize and improve the earlier publications. Two numerical examples are given to show the effectiveness of the obtained results. It is believed that these results are significant and useful for the design and applications of the Cohen–Grossberg neural networks.