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Highlights • A general tridiagonal two-layer neural network model with delay is proposed. • A new method of Hopf bifurcation analysis is introduced by matrix decomposition. • The conditions obtained are simpler than traditional Hurwitz discriminant method. Abstract Firstly, a general tridiagonal two-layer neural network model with 2 n -neuron is proposed, where every layer has time delay. A new method of Hopf bifurcation analysis is introduced by matrix decomposition in this paper. Through factoring the tridiagonal matrix, the characteristic equation of the neural network model is simplified. Secondly, by studying the eigenvalue equations of the related linear system for the special six-neuron (three neurons per layer) two-layer neural network model, the sufficient conditions for experiencing the Hopf bifurcation are obtained. The conditions obtained by the new method proposed in this paper are simpler and more practical than those obtained by the traditional Hurwitz discriminant method. Next, based on the normal form method and the center manifold theorem, the explicit formulae about the stability of the bifurcating periodic solution and the direction of the Hopf bifurcation are established. Finally, the main results obtained in this paper are illustrated by three numerical simulation examples. [ABSTRACT FROM AUTHOR]

This paper considers the Internet congestion control and consensus problems of a multi-agent system composed of a congested node, some aggregation nodes and their following nodes. First, a novel multi-agent Internet congestion control system with both discrete and distributed delays is proposed. Second, the stability and Hopf bifurcation of the congested node are discussed. Subsequently, a simple error controller designed for aggregation nodes, some necessary and sufficient conditions have been established to solve the bifurcation consensus problem between aggregating nodes and the congested node. Bifurcation consensus of aggregation nodes and their following nodes are also obtained through consensus protocols which based on a novel predictive controller. So far, the bifurcation asymptotically consensus of multi-agent Internet congestion control system is achieved. It is worth noting that conditions for global stability of system can be obtained by selecting appropriate control parameters, and conclusions of global asymptotic stability consensus is given to draw conditions for avoiding network congestion in the whole system. Finally, numerical examples illustrated the correctness and validity of the main results. [ABSTRACT FROM AUTHOR]

This paper concerns a model of hierarchical neural network with D n × D n -symmetry and time delays. The linear stability of the steady state is obtained via an artful factorization technique of the characteristic equation corresponding to the linearized system. And we also perform a delay-induced Hopf bifurcation analysis involving spatio-temporal patterns of bifurcated periodic solutions relying on the equivariant Hopf bifurcation theory for retarded functional differential equations and the representation method of Lie groups. Some numerical simulations for a simple example are carried out to demonstrate the application of our theoretical conclusion that multiple periodic orbits could occur simultaneously from Hopf bifurcation in the model. [ABSTRACT FROM AUTHOR]

In this paper, we consider a four-neuron bi-directional associative memory (BAM, for short) neural network with two delays. We choose connection weights and the sum of delays as bifurcation parameters and derive the critical values where a double Hopf bifurcation may occur by analyzing the associated characteristic equation which is a fourth-degree polynomial exponential equation. Meanwhile, we obtain some parameter conditions on the existence of invariant 2-tori of the truncated normal form near the bifurcation point by the center manifold theorem and normal form method. Despite the fact that the higher-degree terms may destroy the invariant 2-tori of the truncated normal form, we prove that the neural network model has quasi-periodic invariant 2-tori for most of the parameter set where the truncated normal form possesses invariant 2-tori in a sufficiently small neighborhood of the bifurcation point. Numerical examples and simulations are given to support the theoretical analysis. [ABSTRACT FROM AUTHOR]

Conduction of action potentials along nerve fiber is an important nonlinear phenomenon related to neural information transition. Recently, the conduction failure behavior that some action potentials with high frequency fail to conduct along C-fiber was identified to be associated with diabetic neuropathy and induced by potassium (K +) channel. In the present paper, the dynamical mechanism for downregulated expression of K + channel induced-reduction of conduction failure, i.e. enhancement of painful information, is acquired with a chain network model composed of Hodgkin-Huxley (HH) neurons. The conduction failure behavior appears for the action potentials induced by stimulation with high frequency and small coupling strength corresponding to C-fiber, and the conduction failure degree reduces with decreasing K + conductance, which closely matches with those observed in the biological experiment. Moreover, the dynamics of the conduction failure behavior are explained with the Hopf bifurcation of HH model. The afterpotential near the Hopf bifurcation exhibits the damping oscillation, which leads to that the current intensity threshold of a pulse stimulation to evoke an action potential from the afterpotential also manifests damping oscillations. Such a current threshold is the intrinsic cause that conduction failure behavior appears for action potential with high frequency. With decreasing K + conductance, the current threshold becomes lower, which is the intrinsic dynamical mechanism for the reduction of conduction failure degree. By using current threshold determined by Hopf bifurcation, the results present the dynamical mechanism of potassium channel induced-conduction failure behavior, which shows that potassium channel is a potential modulation candidate for the pathological pain. [ABSTRACT FROM AUTHOR]

Abstract This paper presents a set of control approaches that can deal with unwanted dynamics such as the Hopf Bifurcation in a Chaotic Associative Memory (CAM) for two distinct goals: the bifurcation critical point shift to anticipate or to postpone the start of a Hopf bifurcation; or to extinguish it. A Washout Filter Control, a Polynomial Control, and an adaptive control for data-driven called the Model-Free Adaptive Control were used in this study. We present the results of numerical simulations in a CAM with three neurons (a system with six dimensions), which were trained classically to retrieve a set of memories. Our analysis includes the dynamics of the bifurcated system, the ability of the network to recover the memories learned and how it is affected by the control strategies. The generality of the proposed control approaches was tested to control a different system and for a different behavior (chaotic dynamics). [ABSTRACT FROM AUTHOR]

Abstract In this paper, a neural networks model with seven neurons and time delay is considered. The model can be described as three triangles sharing one node. The local asymptotical stability of the trivial equilibrium point is studied by analyzing the corresponding characteristic equation. By using the delay as bifurcation parameter, the critical value of bifurcation is given, and then the stability and Hopf bifurcation of the model are discussed. In addition, the stability and bifurcation direction of the bifurcation periodic solution are discussed by using the central manifold theorem and the norm form. The validity of the above theoretical results is verified by numerical simulation. [ABSTRACT FROM AUTHOR]

In this paper, we consider a general 2-neuron network model with reaction–diffusion term and time delay. We study the effect of time delay in kinetic terms of reaction–diffusive system. We mainly investigate the effects of time delay and diffusion term on the stability of the neural network model. Later, we present an algorithm to determine the existence of Hopf bifurcation for the delayed system with reaction–diffusion term along with Neumann boundary conditions. We determine the conditions on the delay parameter for the Hopf bifurcation to exist corresponding to the characteristic equation obtained by linearization of system. At the end, we give some numerical examples along with simulation results to show effectiveness of our analytic findings. [ABSTRACT FROM AUTHOR]

This paper is concerned with the stability and Hopf bifurcation for a fractional-order Susceptible-Vaccinated-Exposed-Infectious-Recovered (SVEIR) computer virus propagation model with a nonlinear incident rate. By employing the linearization technique and Routh-Hurwitz method, the sufficient criterion is established for the locally asymptotic stability of endemic equilibrium point. The Hopf bifurcation is also studied for the SVEIR computer virus model by taking time delay as the bifurcation parameter. The research results show that the stability and Hopf bifurcation of proposed model are significantly affected by both time delay and the order of the fractional derivative. Examples with proper parameters and several simulations are given to illustrate the validity of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, the four-neuron inertial neural system with multiple delays is proposed. By analyzing the associated transcendental characteristic equation, the linear stability of the model is investigated and Hopf bifurcation of the trivial equilibrium point is demonstrated. Periodic solutions bifurcating from the trivial equilibrium point are obtained analytically by using the Perturbation scheme without the normal form method and center manifold theory. Finally, numerical simulations well support the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel fractional-order two-gene regulatory network model with delays, which can describe the memory and hereditary properties of genetic regulatory networks more suitably. It is the first time that the dynamics of the stability and Hopf bifurcation are investigated for the delayed fractional-order model of two-gene regulatory network. The total delay is chosen as the bifurcation parameter of the network, and the sufficient conditions of the stability and Hopf bifurcation are achieved through analyzing its characteristic equation. It is found that the delayed fractional-order genetic network can generate a Hopf bifurcation when the total delay passes through some critical values, which can be determined exactly by dealing with the characteristic equation of the network. Finally, the validity of our theoretical analysis is illustrated by carrying out the numerical simulation for the example, and some desirable dynamical behaviors of the case are obtained by choosing the appropriate fractional order. It is discovered that the onset of the Hopf bifurcation increases distinctly when the fractional order deceases. Therefore, the stability domain of the network is inversely proportion to the fractional order of the network. [ABSTRACT FROM AUTHOR]

In this paper, a love-triangle model with nonlinear romantic sentimental interactions and four time delays is proposed. Regarding time delay as bifurcating parameter, the dynamical behaviors which include local asymptotical stability and Hopf bifurcation are studied in detail by analyzing the characteristic equation corresponding to the linearized system of a love-triangle model. When the delay passes through a sequence of critical values, we find that Hopf bifurcation occurs. Furthermore, stability and direction of bifurcating periodic solution are derived by applying the normal form theory and the center manifold theorem in our love-triangle model. Finally, an illustrative example is also used to support our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, the dynamical behaviors including multistability and bifurcation of a delayed neural network system are investigated. It is shown that the system coexists sixteen stable states with their own domains of attraction. All stable states are determined by using a Lyapunov function. Interestingly, the system exhibits bistability and double cycles with only one equilibrium. The existence of the Hopf bifurcation is well studied. It is shown that the Hopf bifurcation occurs as the time delay reaches a certain value. Simulation experiments supported our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the problem of bifurcation control for a complex networks in a small-world networks model with time delay. By choosing the nonlinear interactive parameter as the bifurcation parameter, we present a Proportional-Derivative (PD) feedback controller to control Hopf bifurcation which inherently happens due to the networks topology. Stability analysis shows that the onset of Hopf bifurcation can be delayed or advanced via a PD controller by setting proper control parameters. Therefore the Hopf bifurcation of the model became controllable to achieve desirable behaviors which is applicable in certain circumstances. Meanwhile, the direction and stability of bifurcating periodic solutions are determined by using the normal form theory and the center manifold theorem. Finally, numerical simulations confirm the effectiveness of the control strategy in controlling the Hopf bifurcation for the complex network model. [ABSTRACT FROM AUTHOR]

A hierarchical network composed of two interacting rings each of which consists of n identical cells with an unidirectional coupling is the topic of this paper. We present a detailed discussion about the linear stability of the equilibrium by analyzing the associated characteristic equation. The local Hopf bifurcation and spatio-temporal patterns of bifurcating periodic oscillations are also given by employing the symmetric Hopf bifurcation theory for delay differential equations. In particular, by using the normal form theory and the center manifold theorem, we derive the formula determining the direction of the Hopf bifurcation and the stability of the bifurcated periodic orbits. An example with numerical simulations is presented to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

This paper investigates the stability and Hopf bifurcation of a Goodwin model with four different delays. Firstly, we present the existence and uniqueness of the positive equilibrium for the system. Then the sum of time delays is chosen as the bifurcation parameter. By analyzing the distribution of characteristic roots of the corresponding linearized system, we obtain the conditions for keeping the system to be stable. Moreover, it is illustrated that the Hopf bifurcation will occur when the delay passes through a critical value. Moreover, some specific formulas for determining the stability and direction of the Hopf bifurcation are obtained by using the normal form theory and the center manifold reduction. Finally, numerical simulation is given to verify the correctness of our theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, a genetic regulatory network with two time delays is considered. Firstly, the stabilities with one delay and two delays are investigated. It is shown that the stability switches for time delays, and Hopf bifurcation may occur within certain range of the model parameters. Then, combining the normal form method and the center manifold theorem, we derive the explicit formulas which determine the direction of the bifurcation and the stability of the bifurcated periodic solutions. Finally, some numerical examples are exploited to support our results. [ABSTRACT FROM AUTHOR]

This paper studies the group consensus problem of second-order multi-agent systems (MASs) with time delays. First, by state transformation method, the group consensus problem of multi-agent systems can be equivalently transformed into the asymptotical stability of a time-delay system. Then, by Lyapunov first method and Hopf bifurcation theory, respectively, we aim to find the upper bound of time delay τ ⁎ such that the multi-agent systems can achieve group consensus for τ ∈ [ 0 , τ ⁎ ) . Finally, simulation examples are given to show the effectiveness of our theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper has two basic objectives: the first is to investigate Hopf Bifurcation in the internal state of a Chaotic Associative Memory (CAM). For a small network with three neurons, resulting in a six-dimensional Equation of State, the existence and stability of Hopf Bifurcation were verified analytically. The second objective is to study how the Hopf Bifurcation changed the external state (output) of CAM, since this network was trained to associate a dataset of input–output patterns. There were three main differences between this study and others: the bifurcation parameter was not a time delay, but a physical parameter of a CAM; the weights of interconnections between chaotic neurons were neither free parameters nor chosen arbitrarily, but determined in the training process of classical AM; the Hopf Bifurcation occurred in the internal state of CAM, and not in the external state (input–output network signal). We present three examples of Hopf Bifurcation: one neuron with supercritical bifurcation while the other two neurons do not bifurcate; two neurons bifurcating into a subcritical bifurcation and one neuron does not bifurcate; and the same example as before, but with a supercritical bifurcation. We show that the presence of a limit cycle in the internal state of CAM prevents output signals from the network converging towards a desired equilibrium state (desired memory), although the CAM is able to access this memory. [ABSTRACT FROM AUTHOR]

In this paper, stability and Hopf bifurcation of reaction–diffusion neural networks with delays is considered, where the sum of the delays can be regarded as a bifurcation parameter. Some sufficient conditions are provided for checking stability and Hopf bifurcation. The particular attention is focused on the change of the stability as the bifurcation parameter τ increased. The computer simulations are provided to verify the efficiency of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a simplified neutral BAM neural network model is considered. Some results of stability and Hopf bifurcations occurring at the zero equilibrium as the delay increases are exhibited. Global existence of periodic solutions is established by using a global Hopf bifurcation result of Wu and a Bendixson criterion for higher dimensional ordinary differential equations due to Li and Muldowney. Finally, some numerical simulations are carried out to support the analytical results. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to analyze the problem of Hopf bifurcation control for a congestion control model in a wireless access network with time delay. By choosing communication delay as a bifurcation parameter, it is shown that when delay passes through a critical value, the Hopf bifurcation occurs. The bifurcation behavior may cause heavy oscillation and induce network instability. In order to control the undesirable Hopf bifurcation, a hybrid control strategy is proposed. By using a linear stability analysis, we show that adjusting the control parameters of the hybrid control strategy properly, the Hopf bifurcation can be delayed or even eliminated completely without changing the equilibrium point of the system. Therefore, this method can effectively control the Hopf bifurcation for the TCP/AQM wireless network model. Theoretical analysis and numerical simulation results show that this method is effective in controlling Hopf bifurcation of congestion control model in wireless network. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the problem of Hopf bifurcation control for a complex network model with time delays is considered by using a new hybrid control strategy, in which state feedback and parameter perturbation are used. To control the Hopf bifurcation, a hybrid control strategy is proposed and the onset of an inherent bifurcation is delayed (advanced) when such a bifurcation is undesired (desired). Furthermore, the dynamic behaviors of the controlled system can also be changed by choosing appropriate control parameters. Numerical simulation results confirm that the new control strategy is efficient in controlling Hopf bifurcation. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a harmonic oscillator with delayed feedback. By studying the distribution of the eigenvalues of the characteristic equation, we drive the critical values where Bogdanov–Takens (B–T) bifurcation and zero-Hopf bifurcation occur. The versal unfoldings of the normal forms at the singularity of B–T and a pure imaginary and a zero eigenvalue singularity are given, respectively. Some numerical simulations verify the theoretical results. [Copyright &y& Elsevier]

Abstract: In this paper we consider an unidirectional ring of n neurons with distributed delays. The effects of the delay, the coupling strengths and the network size on the stability of the system are investigated. Taking the average delay as a bifurcation parameter, we find two critical values depending on the network size and the coupling strengths, at which the system undergoes Hopf bifurcations. By using the method of multiple scales, we can show that these Hopf bifurcating periodic solutions are asymptotically stable. Finally, the theoretical results are illustrated by numerical simulations for the neural networks with three or four neurons. [Copyright &y& Elsevier]

Abstract: In this paper, we deal with a model of single genetic negative feedback autoregulatory system with delay. Choosing the transcriptional rate k as the bifurcation parameter, we demonstrate that the Hopf bifurcation would occur when k exceeds a critical value. By using the normal form theory and center manifold argument, we derive the explicit formulas which determine the bifurcation direction and the stability of bifurcated periodic solution. In order to illustrate our theoretical analysis, a numerical simulation is also included in the end. [Copyright &y& Elsevier]

Abstract: In this paper, a general two-neuron model with distributed delays, self-feedbacks and a weak kernel is studied. It is shown that the Hopf bifurcation occurs as the bifurcation parameter, the mean delay, passes a critical value where a family of periodic solutions emanate from the equilibrium. By applying the frequency domain approach and analyzing the associated characteristic equation, the existence of the bifurcation critical point of the mean delay is determined. The direction and the stability of bifurcating periodic solutions are determined by the Nyquist criterion and the graphical Hopf bifurcation theorem. Numerical simulation results supporting the theoretical analysis are also given. [Copyright &y& Elsevier]