In recent years, with the wide application of artificial neural networks (ANNs) in the field of artificial intelligence, the study of neural network bifurcation dynamics has received much attention and has achieved a large number of results, but the current research only focuses on the binary interactions between neurons and does not take into account the higher-order interactions (HOIs) that exist between neurons. Furthermore, most neural network models focus on ring, star and chain structures, but it is more practical to study multi-layer neural networks. For this reason, this paper develops a three-layer neural network with multiple time delays under HOIs. Firstly, the characteristic equation of network is obtained, and the time delay is selected as the bifurcation parameter. Subsequently, the stability of the neural network and the sufficient condition for the occurrence of Hopf bifurcation are established. Then, the correctness of the theoretical results is verified through numerical simulations. The simulation results demonstrate that the increase of the time delay leads to Hopf bifurcation, which consequently leads to the system oscillation and instability. In addition, the stability of the network is closely related to the higher-order coupling coefficient, self-feedback coefficient and unified connection weight. An increase in the higher-order coupling coefficient expands the stability domain of the network, an increase in the self-feedback coefficient enlarges the stability domain of the network, and an increase in the unified connection weight narrows the stability domain of the network. Finally, our model performs well in terms of function fitting, which also provides some insights into the subsequent modelling and analysis of higher-order neural networks.

The study of dynamics in complex systems has increasingly incorporated higher order interactions, which capture the collective influence among three or more units, extending beyond traditional pairwise connections. Although such interactions are observed in biological neural networks, their precise role in shaping network dynamics and the feasibility of controlling these dynamics remain unclear. This article proposes a controlled diffusion hub neural network model that explicitly includes higher order interactions. To regulate the resulting spatiotemporal dynamics, a cross-node associated delayed feedback control (CNADFC) method is further introduced. Our analysis establishes conditions for local stability, Turing instability, and Hopf bifurcation. We show that while Turing instability cannot arise, spatially periodic patterns emerge under specific parametric conditions. Numerical simulations confirm these theoretical findings and highlight the pronounced effects of self-feedback, control, and first-order interaction on stability and dynamic behaviors; in contrast, higher order interactions exert a comparatively modest influence. Furthermore, simulations illustrate how the CNADFC method can effectively optimize spatiotemporal dynamics. This work advances the understanding of diffusion neural network behavior under complex higher order interaction and provides a reference for the effective control of such networks.

This article studies the multisynchronization of delayed coupled neural networks (DCNNs) with general activation functions (AFs) via impulsive control. At first, a kind of AFs is proposed, and the $\boldsymbol {n}$ -neuron subnetwork with this kind of AFs can produce $\boldsymbol {(p+1)^{n}}$ locally stable equilibrium points (EPs) or periodic orbits (POs) by judging $\boldsymbol {2n(p+1)}$ algebraic inequalities and a nonsingular $\boldsymbol {M}$ -matrix. Compared with some specific AFs, such as the Sigmoid AFs or the saturated AFs, the AFs proposed in this article are more general. In addition, a kind of impulsive controller is designed. Compared with a continuous-time control strategy, the impulsive control strategy proposed in this article can reduce the communication cost and save bandwidth. Moreover, sufficient conditions are given to ensure both dynamical multisynchronization (DMS) and static multisynchronization (SMS) of DCNNs by building the comparison system and using the Lagrange method of variation of parameters. Lastly, an example is illustrated to testify to the validity of the obtained results.

This article investigates the security control problem of Markov jump systems (MJSs) under denial of service (DoS) attacks by constructing a class of discontinuous interval-dependent functional. Motivated by (1) the enhanced practicality and flexibility of multi-sensor sampled-data control with asynchronous packet arrivals and (2) the destabilizing impact of DoS-induced data loss, a resilient asynchronous sampled-data control (RASDC) scheme is designed. RASDC can send data once immediately after the attack, thereby alleviating the performance degradation caused by long-term no input. Furthermore, by utilizing the stability analysis idea of switched systems, a discontinuous interval-dependent functional is constructed, which combines the information of the resilient sampling interval and the attack interval. To overcome the analytical challenges posed by its discontinuities, the discrete-time Lyapunov stability theory, convex combination techniques, and some estimation methods are used to derive the sufficient condition for mean square asymptotic stability. In light of the stability criterion, a design algorithm for the secure controller is provided. Finally, simulation results demonstrate the effectiveness of the proposed approach.

The principal innovative contribution of this study resides in the introduction of a category of fractional delayed large-scale neural networks characterized by intricate topological structures. Additionally, this article provides a comprehensive exploration of novel outcomes linked to fractional order-induced bifurcations in large-scale networks. In the initial step, the correlation of the artificial neural network and the graphical neural network is established through the Mason's diagram method. Subsequently, the system's characteristic equations are derived by employing the Coates' flow graph decomposition method. Moving on, through the concept of the global element, an exhaustive investigation delves into the distribution of eigenroots. The sum of synaptic transmission delays among neurons is considered as a bifurcation parameter, with an analysis focused on the stability of the trivial equilibrium and the existence of the Hopf bifurcation. Following this, the optimal fractional order-dependent stability interval is determined using the implicit function array curve method, presenting a novel approach for critical value determination. Finally, the drawn conclusions are substantiated through multiple sets of computer simulations. It is indicated that an increase in delay precipitates the onset of Hopf bifurcation. Moreover, a reduction in the fractional order significantly improves the steady-state performance of the system. However, once the fractional order value descends below the left stability boundary, the system's stability is compromised, leading to the emergence of periodic oscillations. The prediction algorithm proposed in this article offers valuable insights into selecting the appropriate fractional order for large-scale complex networks.

In recent years, the analysis of the dynamics of annular neural networks has received extensive attention and achieved some achievements. However, most of the current research merely focuses on the single-ring, low-dimension, two rings sharing one neuron cases, without considering the rich coupling modes between rings. In this article, a large-scale time-delay fractional-order dual-loop neural network model with cross-coupling structure is established, in which two rings complete information interaction through two shared neurons. Moreover, the Caputo fractional derivative is introduced in this article to describe the neural network more accurately. First, the transmission time delay between each neuron is selected as the key parameter leading to the bifurcation, and the characteristic equation of the network is creatively derived using the Coates flow graph method. Subsequently, through the holistic element method and magnitude angle formula, we simplify the analytical process. Then, we obtain the stability and Hopf bifurcation criterion of the network. Finally, the conclusions of the theoretical analysis are verified by a series of numerical simulations. The results show that the stability region of the network is closely related to the fractional order, the number of neurons, the distribution of neurons, and the self-feedback coefficients. Moreover, the time delays have a significant effect on the amplitude and period of the Hopf bifurcation.

Spatiotemporal dynamics in the brain have been recognized as strongly related to the formation of perceived and cognitive diseases, such as delusions and hallucinations in Alzheimer's disease. However, two practical considerations are rarely mentioned in related mechanism research: the connectomics networking and the anomalous diffusion generated by the complex medium between neurons and the complex topology of neural networks, respectively. Furthermore, how to optimize the corresponding dynamics behaviors has excellent implications for treating brain diseases. This article first realizes the networking under connectomics for an anomalous-diffusion single-neuron model and applies a nonlinear state feedback control to generate optimized dynamic behaviors, which provides a paradigm of nonequilibrium self-organization driven by anomalous diffusion. Then, by tracing the root distribution of the characteristic equation, some controlled conditions causing or inhibiting Turing instability and Hopf bifurcation are deduced, and the effects of self-diffusion and cross diffusion on Turing instability range are also revealed. At last, thorough numerical simulations are updated to illustrate the results. It is emphasized that delay, self-diffusion, cross diffusion, and fractional order occupy dominant positions in determining the network's spatiotemporal dynamics, and utilizing the control strategy can efficiently reduce Turing instability and delay Hopf bifurcation.

Abstract In this paper, a two-neuron reaction–diffusion neural network with discrete and distributed delays is proposed, and the state feedback control strategy is adopted to achieve control of its spatiotemporal dynamical behaviours. Adding two virtual neurons, the original system is transformed into a neural network only containing the discrete delay. The conditions under which Hopf bifurcation and Turing instability arise are determined through analysis of the characteristic equation. Additionally, the amplitude equations are derived with the aid of weakly nonlinear analysis, and the selection of the Turing patterns is determined. The simulation results demonstrate that the state feedback controller can delay the onset of Hopf bifurcation and suppress the generation of Turing patterns.

Turing patterns in diffusion neural networks are strongly associated with the performance of artificial intelligence model. However, two practical considerations are rarely mentioned in the mechanism research on the problem of diffusion-coupled cellular neural networks (CNNs): the effects of memristor and three-dimensional (3-D) network structure. Furthermore, facilitating the formation of the determined Turing pattern is expected to make the neural network exhibit the desired intelligence. This paper first realizes the 3-D diffusive networking for a primary cell circuit with memristor characteristics and utilizes a proportional-derivative (PD) control strategy to drive the pattern formation. Next, the characteristic equation is derived using the spatial eigenfunction-based decoupling method. Then, by tracing the root distribution of the characteristic equation, some analytical conditions for the stability or forming Turing patterns are deduced. In addition, the central manifold reduction and linear analysis approaches are utilized to derive the amplitude equations of 3-D Turing patterns and investigate their stability. At last, some numerical simulations are provided to illustrate the results. It is also demonstrated that PD control and memristor occupy dominant positions in various CNNs’ Turing patterns. After determining the pattern stability, the implementation of PD control can be considered an effective means to facilitate stable networks to experience Turing instability and generate switchable 3-D pattern.

In this brief, a fractional order proportional-integral-differential neural network (PIDNN) controller based on the beetle swarm optimization algorithm (BSO-PI [Formula: see text]NN) is proposed for multi-input multi-output (MIMO) systems with strong coupling. First, the fractional order PID operator is introduced to the hidden layer neurons of the neural network, where long memory characteristics of the fractional order neurons can improve the control accuracy and convergence speed. Second, a sufficient condition on the learning rate is established to ensure the stability of the controller by the Lyapunov theory. Third, the PI [Formula: see text]NN is initialized by the BSO algorithm to prevent weights from falling into local optima. The proposed fractional order PIDNN controller can eliminate the coupling between variables and achieve desirable control performance without specific system models. To the authors' best knowledge, this is the first work that the fractional order PI [Formula: see text] neurons are employed in neural network. Two simulation examples verify the effectiveness and superiority of the proposed controller.

For decades, studying the dynamic performances of artificial neural networks (ANNs) is widely considered to be a good way to gain a deeper insight into actual neural networks. However, most models of ANNs are focused on a finite number of neurons and a single topology. These studies are inconsistent with actual neural networks composed of thousands of neurons and sophisticated topologies. There is still a discrepancy between theory and practice. In this article, not only a novel construction of a class of delayed neural networks with radial-ring configuration and bidirectional coupling is proposed, but also an effective analytical approach to dynamic performances of large-scale neural networks with a cluster of topologies is developed. First, Coates' flow diagram is applied to acquire the characteristic equation of the system, which contains multiple exponential terms. Second, by means of the idea of the holistic element, the sum of the neuron synapse transmission delays is regarded as the bifurcation argument to investigate the stability of the zero equilibrium point and the beingness of Hopf bifurcation. Finally, multiple sets of computerized simulations are utilized to confirm the conclusions. The simulation results expound that the increase in transmission delay may cause a leading impact on the generation of Hopf bifurcation. Meanwhile, the number and the self-feedback coefficient of neurons are also playing significant roles in the appearance of periodic oscillations.

In this article, the heterogeneity of the electromagnetic field is taken into account and thus the diffusion effect is introduced into the artificial neural network modeling. The first attempt of a class of large-scale bidirectional associative memory neural networks is provided, incorporating diffusion and delays. Using Coates’ flow diagram is able to efficiently and accurately capture the characteristic equations of large-scale reaction-diffusion neural networks. Furthermore, by tracing the distribution of characteristic roots driven by the time delay, a criterion on the local stability is determined and the critical tipping point caused by Hopf bifurcation is also predicted, respectively. Numerical simulations are eventually conducted to demonstrate the practical implications of the theory. It is shown that spatiotemporal dynamic behaviors of neural networks suggested are significantly affected by the transmission delay, the system scale, the self-feedback coefficient and the diffusivity.

Real neural networks are characterized by large-scale and complex topology. However, the current dynamical analysis is limited to low-dimensional models with simplified topology. Therefore, there is still a huge gap between neural network theory and its application. This article proposes a class of large-scale neural networks with a ring-hub structure, where a hub node is connected to n peripheral nodes and these peripheral nodes are linked by a ring. In particular, there exists a hybrid coupling mode in the network topology. The mathematical model of such systems is described by fractional-order delayed differential equations. The aim of this article is to investigate the local stability and Hopf bifurcation of this high-dimensional neural network. First, the Coates flow graph is employed to obtain the characteristic equation of the linearized high-dimensional neural network model, which is a transcendental equation including multiple exponential items. Then, the sufficient conditions ensuring the stability of equilibrium and the existence of Hopf bifurcation are achieved by taking time delay as a bifurcation parameter. Finally, some numerical examples are given to support the theoretical results. It is revealed that the increasing time delay can effectively induce the occurrence of periodic oscillation. Moreover, the fractional order, the self-feedback coefficient, and the number of neurons also have effects on the onset of Hopf bifurcation.

A large number of experiments have proved that the ring structure is a common phenomenon in neural networks. Nevertheless, a few works have been devoted to studying the neurodynamics of networks with only one ring. Little is known about the dynamics of neural networks with multiple rings. Consequently, the study of neural networks with multiring structure is of more practical significance. In this article, a class of high-dimensional neural networks with three rings and multiple delays is proposed. Such network has an asymmetric structure, which entails that each ring has a different number of neurons. Simultaneously, three rings share a common node. Selecting the time delay as the bifurcation parameter, the stability switches are ascertained and the sufficient condition of Hopf bifurcation is derived. It is further revealed that both the number of neurons in the ring and the total number of neurons have obvious influences on the stability and bifurcation of the neural network. Ultimately, some numerical simulations are given to illustrate our qualitative results and to underpin the discussion.

The dynamics of neural networks has been widely concerned by scholars. However, most of the previous results on dynamical bifurcations are limited to few nodes coupling neural networks which modeled by differential equations with integer-order derivative, and few efforts have been contributed to studying the bifurcation behaviors of large-scale fractional-order neural networks. Furthermore, the structural characteristics of networks are also of great research value. Among them, the ring structure is a common phenomenon in neural networks. Although there are few papers on the bifurcation analysis of ring-structured neural networks recently, they consider only the case of a single ring. In this article, the dynamical analysis and design for a class of large-scale-delayed fractional-order neural networks with multiple rings and hub structure are investigated. First, the time delay is considered to be the bifurcation parameter and the formula of Coates’ flow graph is adopted to obtain the characteristic equation of large-scale networks. Second, by analyzing the complex radial and circular connections of neurons, the delay-induced Hopf bifurcation sufficient conditions for the neural network are established. Finally, the theoretical results are substantiated by a number of numerical simulation experiments and the relationships between the onset of bifurcation and the fractional order, the number of neurons, and the number of rings are revealed.

Recently, the dynamics of delayed neural networks has always incurred the widespread concern of scholars. However, they are mostly confined to some simplified neural networks, which are only made up of a small amount of neurons. The main cause is that it is difficult to decompose and analyze generally high-dimensional characteristic matrices. In this article, for the first time, we can solve the computing issues of high-dimensional eigenmatrix by employing the formula of Coates flow graph, and the dynamics is considered for a bidirectional neural network with super-ring structure and multiple delays. Under certain circumstances, the characteristic equation of the linearized network can be transformed into the equation with integration element. By analyzing the equation, we find that the self-feedback coefficient and the delays have significant effects on the stability and Hopf bifurcation of the network. Then, we achieve some sufficient conditions of the stability and Hopf bifurcation on the network. Furthermore, the obtained conclusions are applied to design a standardized high-dimensional network with bidirectional ring structure, and the scale of the standardized high-dimensional network can be easily extended or reduced. Afterward, we propose some designing schemes to expand and reduce the dimension of the standardized high-dimensional network. Finally, the results of theories are coincident with that of experiments.

This article focuses on the hybrid effects of memristor characteristics, time delay, and biochemical parameters on neural networks. First, we propose a novel neuron system with memristor and time delays in which the memristor is characterized by a smooth continuous cubic function. Second, the existence of equilibria of this type of neuron system is examined in the parameter space. Sufficient conditions that ensure the stability of equilibria and occurrence of pitchfork bifurcation are given for the memristor-based neuron system without delay. Third, some novel criteria of the addressed neuron system are constructed for guaranteeing the delay-dependent and delay-independent stability. The specific conditions are provided for Hopf bifurcations, and the properties of Hopf bifurcation are ascertained using the center manifold reduction and the normal form theory. Moreover, there exists a phenomenon of bistability for the delayed memristor-based neuron system having three equilibria. Finally, the effectiveness of the theoretical results is demonstrated by numerical examples.

Bifurcation control remains largely unresolved for fractional-order dynamical systems. This article addresses the optimal control issue of bifurcation for a delayed complex networks model with Caputo derivative, where the time delay is selected as the variable parameter. We first devise a fractional-order proportional-integral-derivative (PID) feedback synthesis for regulation of the Hopf bifurcation embedded in a delayed small-world network model with Caputo derivative. Dynamic stability criterion and Hopf bifurcation condition are obtained by carrying out the eigenvalue analysis of the controlled network. The stability range of the parameters of the PID control is evaluated completely for the small-world network. We can optimize the dynamics of stability and bifurcation of small-world networks by manipulating the gain parameters. Finally, we implement some simulations to show the performance of the presented PID scheme. The numerical simulations verify the advantage of the fractional PID controller in bifurcation control.

This paper is concerned with leader-following synchronization in a heterogeneous network coupled by a group of heterogeneous LC oscillators. Both the dynamics between the leader and each follower and the dynamics among the followers are nonidentical. A sampled-data-based protocol is proposed. The sufficient criteria for quasi-synchronization are derived based on the delayed input approach and the Lyapunov stability method. Furthermore, the quasi-synchronization error bound can be solved. Finally, simulations are given to illustrate the theoretical results.

It has been demonstrated in a large number of experimental results that small RNAs (sRNAs) play a vital role in gene regulation processes. Thus, the gene regulation process is dominated by sRNAs in addition to messenger RNAs and proteins. However, the regulation mechanism of sRNAs is not well understood and there are few models considering the effect of sRNAs. So it is of realistic biological background to include sRNAs when modeling gene networks. In this paper, sRNAs are incorporated into the process of gene expression and a new differential equation model is put forward to describe cyclic genetic regulatory networks with sRNAs and multiple delays. We mainly investigate the stability and bifurcation criteria for two cases: 1) positive cyclic genetic regulatory networks and 2) negative cyclic genetic regulatory networks. For a positive cyclic genetic regulatory network, it is revealed that there may exist more than one equilibrium and the multistability can appear. Sufficient conditions are established for the delay-independent stability and fold bifurcations. It is found that the dynamics of positive cyclic gene networks has no bearing on time delays, but depends on the biochemical parameters, the Hill coefficient and the equilibrium itself. For a negative cyclic genetic regulatory network, it is proved that there exists a unique equilibrium. Delay-dependent conditions for the stability are derived, and the existence of Hopf bifurcations is examined. Different from the delay-independent stability of positive gain networks, the stability of equilibrium is determined not only by the biochemical parameters, the Hill coefficient and the equilibrium itself, but also by the total delay. At last, three illustrative examples are provided to validate the major results.

In this paper, we consider the problem of bifurcation control for a single genetic regulatory network model with time delay. The time delay is selected as the bifurcation parameter of it. We control the network by using a Proportional-Integral-Derivative (PID) feedback controller. It is the first time the feasible region of the parameters of the controller is provided for the network structure remains the same. Afterwards, we perform a stability analysis for the characteristic equation of the network. It is found that the onset of Hopf bifurcation can be changed by changing the parameters of the controller. Finally, the relationship is obtained between the stability region and the parameters of the controller through numerical simulaing for a specific example.

This paper studies the stability and Hopf bifurcation in a class of high-dimension neural network involving the discrete and distributed delays under a new framework. By introducing some virtual neurons to the original system, the impact of distributed delay can be described in a simplified way via an equivalent new model. This paper extends the existing works on neural networks to high-dimension cases, which is much closer to complex and real neural networks. Here, we first analyze the Hopf bifurcation in this special class of high dimensional model with weak delay kernel from two aspects: one is induced by the time delay, the other is induced by a rate parameter, to reveal the roles of discrete and distributed delays on stability and bifurcation. Sufficient conditions for keeping the original system to be stable, and undergoing the Hopf bifurcation are obtained. Besides, this new framework can also apply to deal with the case of the strong delay kernel and corresponding analysis for different dynamical behaviors is provided. Finally, the simulation results are presented to justify the validity of our theoretical analysis.

In this paper, a fractional-order recurrent neural network is proposed and several topics related to the dynamics of such a network are investigated, such as the stability, Hopf bifurcations, and undamped oscillations. The stability domain of the trivial steady state is completely characterized with respect to network parameters and orders of the commensurate-order neural network. Based on the stability analysis, the critical values of the fractional order are identified, where Hopf bifurcations occur and a family of oscillations bifurcate from the trivial steady state. Then, the parametric range of undamped oscillations is also estimated and the frequency and amplitude of oscillations are determined analytically and numerically for such commensurate-order networks. Meanwhile, it is shown that the incommensurate-order neural network can also exhibit a Hopf bifurcation as the network parameter passes through a critical value which can be determined exactly. The frequency and amplitude of bifurcated oscillations are determined.

This paper studies the stability and Hopf bifurcation in a high-dimension neural network involving the discrete and distributed delays. Such model extends the existing models of neural networks from low-dimension to high-dimension. Therefore, our model is much close to large real neural networks. Here, the delay is chosen as the bifurcation parameter and we obtain the sufficient conditions for the system keeping stable and undergoing the Hopf bifurcation. Moreover, the software package DDE-BIFTOOL is introduced to better display the properties of the system and the effect of gain parameters of the system and delay kernel on the onset of the bifurcation. The simulation results further justify the validity of our theoretical analysis.

This paper studies oscillatory dynamics of two-gene regulatory networks which are mediated by small RNAs (sRNAs) and subject to multiple delays. First, stability of the positive fixed point and the existence of the local Hopf bifurcation are examined for sRNAs-mediated two-gene regulatory networks. Then sufficient conditions for periodic oscillation are established for such networks with multiple delays. Computer simulations are presented to illustrate the proposed results.

This paper is concerned with the bifurcation problem of bidirectional associative memory (BAM) neural networks with two delays. Two delays play different roles in dynamical behaviors of BAM neural networks. The characteristic equation associated with delayed BAM neural networks is analyzed, which gives the distribution of the eigenvalues. Then some important dynamic properties of the trivial steady state, such as the local stability and the existence of Hopf bifurcation, are obtained. The theoretical results are further validated by numerical simulations.

Recent studies on Hopf bifurcations of neural networks with delays are confined to simplified neural network models consisting of only two, three, four, five, or six neurons. It is well known that neural networks are complex and large-scale nonlinear dynamical systems, so the dynamics of the delayed neural networks are very rich and complicated. Although discussing the dynamics of networks with a few neurons may help us to understand large-scale networks, there are inevitably some complicated problems that may be overlooked if simplified networks are carried over to large-scale networks. In this paper, a general delayed bidirectional associative memory neural network model with n + 1 neurons is considered. By analyzing the associated characteristic equation, the local stability of the trivial steady state is examined, and then the existence of the Hopf bifurcation at the trivial steady state is established. By applying the normal form theory and the center manifold reduction, explicit formulae are derived to determine the direction and stability of the bifurcating periodic solution. Furthermore, the paper highlights situations where the Hopf bifurcations are particularly critical, in the sense that the amplitude and the period of oscillations are very sensitive to errors due to tolerances in the implementation of neuron interconnections. It is shown that the sensitivity is crucially dependent on the delay and also significantly influenced by the feature of the number of neurons. Numerical simulations are carried out to illustrate the main results.

In this paper, we introduce the fractional order into a three-node recurrent neural network model, and then consider the effect of the order on the system dynamics for the neural network based on a fractional-order differential equation. By applying the existing theorems on the stability of commensurate fractional-order systems, we investigate the linear stability and Hopf-type bifurcation for the fractional-order neural network model. Our analysis shows that the equilibrium point, which is unstable in the classic integer-order model, can become asymptotically stable in our fractional-order model, which is also confirmed by numerical simulations. Moreover, we also present simulation results of limit cycles produced by the fractional-order neural network model. It is shown that the amplitude of limit cycles increases with the order, while the frequency of limit cycles has robustness against the change in the order due to its small variation.

Various local periodic solutions may represent different classes of storage patterns or memory patterns, and arise from the different equilibrium points of neural networks (NNs) by applying Hopf bifurcation technique. In this paper, a bidirectional associative memory NN with four neurons and multiple delays is considered. By applying the normal form theory and the center manifold theorem, analysis of its linear stability and Hopf bifurcation is performed. An algorithm is worked out for determining the direction and stability of the bifurcated periodic solutions. Numerical simulation results supporting the theoretical analysis are also given.