Stability and Hopf-Bifurcating Periodic Solution for Delayed Hopfield Neural Networks with n Neuron

We consider a system of delay differential equations which represents the general model of a Hopfield neural networks type. We construct some new sufficient conditions for local asymptotic stability about the trivial equilibriumbased on the connectionweights and delays of the neural system.We also investigate the occurrence of an Andronov-Hopf bifurcation about the trivial equilibrium. Finally, the simulating results demonstrate the validity and feasibility of our theoretical results.


Introduction
Analysis of neural networks from the viewpoint of nonlinear dynamics is helpful in solving problems of theoretical and practical importance.The Hopfield neural networks (HNNs) have diverse applications in many areas such as classification, associative memory, pattern recognition, parallel computations, and optimization [1][2][3][4][5].These vast applications have been the focus of detailed studies by researchers.So, we believe that the study of these neural networks models is important.Actually, in some artificial neural network applications such as content-addressable memories, information is stored as stable equilibrium points of the system.Retrieval occurs when the system is initialized within the basin of attraction of one of the equilibria and the network is allowed to be stabilized in its steady state.Time delay has important influences on the dynamical behavior of neural networks.Marcus and Westervelt [6] first considered the effect of including discrete time delays in the connection terms to represent the time of propagation between neurons.They found out that the delay can destabilize the network as a whole and create oscillatory behavior.
The study of the local asymptotic stability and Andronov-Hopf bifurcations of neural network models with multiple time delays are complex.In order to reach a deep and clear understanding of the dynamics of such models, most researchers have limited their study to the models with a single delay [7,8].In some papers, multiple delays are considered but there are no self-connection terms and moreover the systems with two delays have been generally investigated [9][10][11].For example, Liao et al. [10] investigated the stability of a two-neuron system with different time delay as follows: They showed that (0, 0) is a unique fixed point of the mentioned system if  1  2 (1 −  1 )(1 −  2 ) ≤ 1.They estimated the length of delays for which local asymptotical stability is preserved.So, they achieved a delay-dependent stability condition with delayed system.As another example, Olien and Bélair [8] investigated a system with two delays; that is, (  ( −   )) ,  = 1, 2. ( They discussed several cases, such as  1 =  2 and  11 =  22 = 0.They obtained some sufficient conditions for the stability of the stationary point (0, 0) of the latest system and showed that this system may undergo some bifurcations at certain values of the parameters.Songa et al. investigated the stability and Hopf bifurcation in an unidirectional ring of  neurons [12] but the model considered here is more general than the one in Song's studies.In fact, we have considered a Hopfield neural network with arbitrary neurons in which each neuron is bidirectionally connected to all the others.The Lyapunov stability theorem is used to establish the sufficient condition for the asymptotic stability of the equilibrium point in recent studies but, here, we obtain sufficient conditions for local asymptotic stability based on analyzing the associated characteristic transcendental equation.In this paper, delayindependent and delay-dependent sufficient conditions for local asymptotic stability are obtained and the Andronov-Hopf bifurcation for delayed Hopfield neural networks with n neuron is studied.

Local Analysis of a Neural Network with Delays
Consider the following delayed neural network described as: where   () represents the activation state of th neuron ( = 1, 2, . . ., ) at time ,   is the weight of synaptic connections from th neuron to th neuron, and   ≥ 0 is the time delay.
Proof.We suppose that  =  +  is a root of characteristic equation (13). is a root of ( 13) if and only if  satisfies P (, ) = P ( + , ) Therefore, there must exist some  (1 ≤  ≤ ), such that Let   (, ) and   (, ) be the real and imaginary parts of (16), respectively; we have For proving the first part of the theorem, suppose that  ≥ 0; then, similar to the proof of the theorems of Gupta et al. [13], we can prove that |  (, )| > 0. Thus, we have demonstrated that if |  (, )| = 0, then  < 0. It completes the proof.
For proving the sccond part of the theorem, suppose that matrix  is a Hurwitz matrix (i.e.,   < 0).

Boundedness
Proof.Let  : R → R  be the solution of the initial value problem In consequence, Corollary 6.2 of Chapter I in [14] implies that The lower bound can be verified analogously.(43)

Bifurcation Analysis
Mathematical model is generally the first approximation of any considered real systems.More realistic models should include some of the earlier states of the system; that is, the model should include time delay.In this section, we will consider the effect of the time delay involved in the feedback control.The main attention here will be focused on Andronov-Hopf bifurcation.We know that the number of the eigenvalues of the characteristic equation ( 13) with negative real parts, counting multiplicities, can change only when the eigenvalues become purely imaginary pairs as the time delay  and the components of  are varied.Note that these components are independent of the delay.As seen in ( 13), when   ̸ = 1,  = 1, 2, . . ., , none of the roots of P(, ) is zero.Thus, the trivial equilibrium  = 0 becomes unstable only when (13) has at least a pair of purely imaginary roots ± ( is the imaginary unit) at which an Andronov-Hopf bifurcation occurs.We will determine if the solution curve of the characteristic equation ( 13) crosses the imaginary axis.We regard the time delay  as the parameter for considering the Hopf-bifurcation aspect of the trivial equilibrium of the system (3).
Proof.In the first case, setting the real and imaginary parts of (45) to zero, we have Taking squares and adding the two above equations, we have We denote the positive root by  * = √ 2  − 1.The unique solution  =  ∈ [0, 2] of ( 47) is  =  = arccos(1/  ) since sin() = (−/  ) > 0. Therefore, for the imaginary root  =  of (45), we have a sequence {  } ∞ 0 as follows: So,  Re()/ is positive at  =  * .Thus, the solution curve of the characteristic equation (45) crosses the imaginary axis.This shows that an Andronov-Hopf bifurcation occurs at  =  * > 0. When  <  * , the origin of state space of system (3) is locally asymptotically stable by continuity.Note that if   > 1, then (45) has a real root  > 0 and, in this case, the origin of the state space of system (3) is unstable.On the other hand, as a result of the abovementioned formulas, we can say that the characteristic equation ( 21) with condition (i) has a simple pair of purely imaginary roots ± * at each   ,  = 0, 1, 2, . .., where   was presented above.

Numerical Simulation
For the numerical simulation, a program has been developed in Matlab.
With the abovementioned values of the parameters, Figure 3 is obtained.
With the abovementioned values of the parameters, Figure 4 is obtained.

Conclusions
In this paper, we have studied the stability and numerical solutions of a Hopfield delayed neural network system which is more general than the models applied by earlier researchers.In fact, our focus here is on a Hopfield neural network with arbitrary neurons in which each neuron is bidirectionally connected to all the others.By analyzing the associated characteristic transcendental equation, some delay-dependent and delay-independent conditions, which can easily be examined, were established to guarantee the origins of the state space of the model to have local asymptotical stability.Since the characteristic equation of the linearized system at the zero solution involves exponential functions, it is too difficult to investigate the conditions under which the entire characteristic roots have negative real parts.Here, we have reached conditions under which the stability of a matrix consisting of the coefficient of system guarantees the asymptotic stability of the origin of the state space of the network.Also, by considering the feasibility and ease of analyzing the stability of the aforementioned matrix, our approach can be considered as highly practical.Furthermore, we have investigated the occurrence of the Andronov-Hopf bifurcation in the above stated system.Simulation examples have been employed to illustrate the theories.Motivated by the novel method proposed by Leonov and Kuznetsov [15] about finding hidden attractors exploited for nonlinear dynamic systems, we plan to employ their ideas and findings in our upcoming surveys on the Hopfield neural network.
To this end, Lyapunov values will undoubtedly play a very practical and significant role.

Corollary 5 .
If  : R → R is a continuous and bounded map with sup ∈R |()| ≤  in addition and  : R → R is a periodic solution of (3) so that 0 is in the range of , then max ∈R ‖ ()‖ 2 ≤  √