Stability and Bifurcation Analysis for a Class of Generalized Reaction-Diffusion Neural Networks with Time Delay

Abstract

Considering the fact that results for static neural networks are much more scare than results for local field neural networks and our purpose letting the problem researched be more general in many aspects, in this paper, a generalized neural networks model which includes reaction-diffusion local field neural networks and reaction-diffusion static neural networks is built and the stability and bifurcation problems for it are investigated under Neumann boundary conditions. First, by discussing the corresponding characteristic equations, the local stability of the trivial uniform steady state is discussed and the existence of Hopf bifurcations is shown. By using the normal form theory and the center manifold reduction of partial function differential equations, explicit formulae which determine the direction and stability of bifurcating periodic solutions are acquired. Finally, numerical simulations show the results.

Academic Editor:Carlo Bianca

Institute of Applied Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, China

Received 5 July 2015; Revised 3 August 2015; Accepted 10 August 2015; 19 January 2016

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

In the past several decades, the dynamics of neural networks have been extensively investigated.

The artificial neural network has been used widely in various fields such as signal processing, pattern recognition, optimization, associative memories, automatic control engineering, artificial intelligence, and fault diagnosis, because it has the characteristics of self-adaption, self-organization, and self-learning.

Most of the phenomena occurring in real-world complex systems do not have an immediate effect but appear with some delay; for example, there exist time delays in the information processing of neurons. Therefore, time delays have been inserted into mathematical models and in particular in models of the applied sciences based on ordinary differential equations. The delayed axonal signal transmissions in the neural network models make the dynamical behaviors become more complicated, because a time delay into an ordinary differential equation could change the stability of the equilibrium (stable equilibrium becomes unstable) and could cause fluctuations, and Hopf bifurcation can occur (see [1]). And in [1] we can know the time delays' effects from the work by Carlo Bianca, Massimiliano Ferrara, and Luca Guerrini. So, the delay is an important control parameter.

In addition, we must consider that the activations vary in space as well as in time, because the electrons move in asymmetric electromagnetic fields, and there exists diffusion in neural network (see [2]).

In the past, the main work was to research local field neural networks, and static neural networks were rarely studied. Considering the fact that the problem of generalized neural network is more general in many aspects; in this paper, we will investigate a class of generalized neural networks which combine local field neural networks and static neural networks.

In order to study the effect of time delays and diffusion on the dynamics of a neural network model, in [3], Gan and Xu considered the following neural network model: [figure omitted; refer to PDF]

Motivated by the works of Gan and Xu, in this paper, we are concerned with the following neural network system with time delay and reaction-diffusion: [figure omitted; refer to PDF] with initial and boundary conditions (Neumann boundary conditions): [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are random constants, where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] represent the neuron charging time constants, [figure omitted; refer to PDF] represents the signal transmission time delay, [figure omitted; refer to PDF] and [figure omitted; refer to PDF] represent the smooth diffusion operators, [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] represent connecting weight coefficients, and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] represent the coefficients of [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , respectively. [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are the state variables and space variable, respectively. [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the action functions of the neurons satisfying [figure omitted; refer to PDF] . [figure omitted; refer to PDF] is a bounded domain in [figure omitted; refer to PDF] with smooth boundary [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] denotes the outward normal derivative on [figure omitted; refer to PDF] .

The organization of this paper is as follows. In Section 2, by analyzing the corresponding characteristic equations, we discuss the local stability of trivial uniform steady state and the existence of Hopf bifurcations of (2) and (3). In Section 3, by applying the normal form and the center manifold theorem, closed-form expressions are derived which allow us to determine the direction of the Hopf bifurcations and the stability of the periodic solutions in (2) and (3) (see [2]). In Section 4, numerical simulations are carried out to illustrate the main theoretical results.

2. Local Stability and Hopf Bifurcation

Obviously, we can easily show that system (2) always has a trivial uniform steady state [figure omitted; refer to PDF] .

Here, we use [figure omitted; refer to PDF] as the eigenvalues of the operator [figure omitted; refer to PDF] on [figure omitted; refer to PDF] with the homogeneous Neumann boundary conditions and [figure omitted; refer to PDF] as the eigenspace corresponding to [figure omitted; refer to PDF] in [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] , let [figure omitted; refer to PDF] be an orthonormal basis of [figure omitted; refer to PDF] , and let [figure omitted; refer to PDF] . Then, [figure omitted; refer to PDF]

Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , where [figure omitted; refer to PDF]

First, we linearize system (2) at [figure omitted; refer to PDF] . Then, [figure omitted; refer to PDF] . [figure omitted; refer to PDF] is invariant under the operator [figure omitted; refer to PDF] for each [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is an eigenvalue of [figure omitted; refer to PDF] if and only if it is an eigenvalue of the matrix [figure omitted; refer to PDF] for some [figure omitted; refer to PDF] , in which case, there is an eigenvalue in [figure omitted; refer to PDF] .

The characteristic equation of [figure omitted; refer to PDF] is of the form [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Letting [figure omitted; refer to PDF] , then (6) becomes [figure omitted; refer to PDF]

Obviously, [figure omitted; refer to PDF]

Obviously, if the following holds:

[figure omitted; refer to PDF] : [figure omitted; refer to PDF]

then [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Hence, if [figure omitted; refer to PDF] holds, when [figure omitted; refer to PDF] , the trivial uniform steady state [figure omitted; refer to PDF] of problems (2) and (3) is locally stable.

Let [figure omitted; refer to PDF] be a solution of (6), separating real and imaginary parts; then, we can get that [figure omitted; refer to PDF]

Squaring and adding the two equations of (11), we obtain that [figure omitted; refer to PDF]

Letting [figure omitted; refer to PDF] , then (12) becomes [figure omitted; refer to PDF]

Obviously, it is easy to calculate that [figure omitted; refer to PDF]

Let [figure omitted; refer to PDF]

Therefore, if [figure omitted; refer to PDF] , (13) has no positive roots. Then, if [figure omitted; refer to PDF] and [figure omitted; refer to PDF] Holds, the trivial uniform steady state [figure omitted; refer to PDF] of system (2) is locally asymptotically stable for all [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

For [figure omitted; refer to PDF] , if [figure omitted; refer to PDF] , then (12) has a unique positive root [figure omitted; refer to PDF] , where [figure omitted; refer to PDF]

It means that the characteristic equation (6) admits a pair of purely imaginary roots of the form [figure omitted; refer to PDF] for [figure omitted; refer to PDF] .

Take [figure omitted; refer to PDF] . Obviously, (12) holds if and only if [figure omitted; refer to PDF] . Now, we define that [figure omitted; refer to PDF]

Then, for [figure omitted; refer to PDF] , when [figure omitted; refer to PDF] , (6) has a pair of purely imaginary roots [figure omitted; refer to PDF] and all roots of it have negative real parts for [figure omitted; refer to PDF] . It is easy to see that if [figure omitted; refer to PDF] holds, the trivial uniform steady state [figure omitted; refer to PDF] is locally stable for [figure omitted; refer to PDF] . Hence, on the basis of the general theory on characteristic equations of delay-differential equations from [3, Theorem 4.1], we can know that [figure omitted; refer to PDF] remains stable when [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] .

Now, we claim that [figure omitted; refer to PDF]

This will mean that there exists at least one eigenvalue with positive real part when [figure omitted; refer to PDF] . In addition, the conditions for the existence of a Hopf bifurcation [2] are then satisfied generating a periodic solution. To this end, we differentiate (6) about [figure omitted; refer to PDF] ; then, [figure omitted; refer to PDF]

So, we know that [figure omitted; refer to PDF]

Therefore, [figure omitted; refer to PDF] [figure omitted; refer to PDF]

By (11), we can obtain that [figure omitted; refer to PDF]

Because [figure omitted; refer to PDF] , so [figure omitted; refer to PDF]

Hence, the transversal condition holds and a Hopf bifurcation occurs when [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

Consequently, we gain the following results.

Theorem 1.

Let [figure omitted; refer to PDF] and let [figure omitted; refer to PDF] be defined by (15). For system (2), let [figure omitted; refer to PDF] hold. If [figure omitted; refer to PDF] , the trivial uniform steady state [figure omitted; refer to PDF] of system (2) is locally asymptotically stable when [figure omitted; refer to PDF] ; if [figure omitted; refer to PDF] , the trivial uniform steady state [figure omitted; refer to PDF] is asymptotically stable for [figure omitted; refer to PDF] and is unstable for [figure omitted; refer to PDF] ; furthermore, system (2) undergoes a Hopf bifurcation at [figure omitted; refer to PDF] when [figure omitted; refer to PDF] .

3. Direction and Stability of Hopf Bifurcation

In Section 2, we have demonstrated that systems (2) and (3) undergo a train of periodic solutions bifurcating from the trivial uniform steady state [figure omitted; refer to PDF] at the critical value of [figure omitted; refer to PDF] . In this section, we derive explicit formulae to determine the properties of the Hopf bifurcation at critical value [figure omitted; refer to PDF] by using the normal form theory and center manifold reduction for PFDEs. In this section, we also let the condition [figure omitted; refer to PDF] hold and [figure omitted; refer to PDF] . And the work of Bianca and Guerrini in papers [4-7] is the founder of the method in this section.

Set [figure omitted; refer to PDF] . We first should normalize the delay [figure omitted; refer to PDF] by the time-scaling [figure omitted; refer to PDF] . Then, (2) can be rewritten in the fixed phase space [figure omitted; refer to PDF] as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is defined by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] .

By the discussion in Section 2, we can know that the origin [figure omitted; refer to PDF] is a steady state of (24) and [figure omitted; refer to PDF] are a pair of simple purely imaginary eigenvalues of the linear equation [figure omitted; refer to PDF] and the functional differential equation [figure omitted; refer to PDF]

On the basis of the Riesz representation theorem, there exists a function [figure omitted; refer to PDF] of bounded variation for [figure omitted; refer to PDF] such that [figure omitted; refer to PDF]

Here, we choose that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the Dirac delta function.

Let [figure omitted; refer to PDF] denote the infinitesimal generator of the semigroup induced by the solutions of (27) and let [figure omitted; refer to PDF] be the formal adjoint of [figure omitted; refer to PDF] under the bilinear pairing [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Then, [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are a pair of adjoint operators.

By the discussions in Section 2, we can realize that [figure omitted; refer to PDF] has a pair of simple purely imaginary eigenvalues [figure omitted; refer to PDF] and they are also eigenvalues of [figure omitted; refer to PDF] since [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are adjoint operators. Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] be the center spaces of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] associated with [figure omitted; refer to PDF] , respectively. Hence, [figure omitted; refer to PDF] is the adjoint space of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

Let [figure omitted; refer to PDF] then, [figure omitted; refer to PDF] is a basis of [figure omitted; refer to PDF] associated with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is a basis of [figure omitted; refer to PDF] associated with [figure omitted; refer to PDF] .

Let [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] for [figure omitted; refer to PDF] , and let [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] for [figure omitted; refer to PDF] .

Now we define that [figure omitted; refer to PDF] , and construct a new basis [figure omitted; refer to PDF] for [figure omitted; refer to PDF] by [figure omitted; refer to PDF]

Hence, [figure omitted; refer to PDF] , which is the second-order identity matrix. Moreover, we define [figure omitted; refer to PDF] for [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for [figure omitted; refer to PDF] . Then, the center space of linear equation (26) is given by [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] denotes the complementary subspace of [figure omitted; refer to PDF] , where [figure omitted; refer to PDF]

Let [figure omitted; refer to PDF] be defined by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is given by [figure omitted; refer to PDF]

Then, we have rewritten system (24), and it can be rewritten as follows: [figure omitted; refer to PDF]

The solution of (24) on the center manifold is given by [figure omitted; refer to PDF]

Letting [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

We can use some easy computations to show that [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Setting [figure omitted; refer to PDF] , by calculating, we get that [figure omitted; refer to PDF]

Because there are [figure omitted; refer to PDF] and [figure omitted; refer to PDF] in [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , we still need to compute them.

By [4], we know that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] for [figure omitted; refer to PDF] , with [figure omitted; refer to PDF] . It follows from (43), (48), and (49) that [figure omitted; refer to PDF]

By (49), we have that for [figure omitted; refer to PDF] [figure omitted; refer to PDF]

Comparing the coefficients with (49), we get that for [figure omitted; refer to PDF] [figure omitted; refer to PDF]

By (50), (52), and the definition of [figure omitted; refer to PDF] , we get that [figure omitted; refer to PDF]

Noticing that [figure omitted; refer to PDF] , hence, [figure omitted; refer to PDF] where [figure omitted; refer to PDF] which is a constant vector.

In a similar way, by (50) and (53), we have that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] which is also a constant vector.

In what follows, we seek appropriate [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . From the definition of [figure omitted; refer to PDF] and (50), we can obtain that [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Substituting (55) and (59) into (57), we can obtain that [figure omitted; refer to PDF]

In a similar way, substituting (56) and (60) into (58), we obtain that [figure omitted; refer to PDF]

Therefore, we can compute the following values: [figure omitted; refer to PDF] which determine the quantities of bifurcating periodic solutions in the center manifold at the critical value [figure omitted; refer to PDF] ; that is, [figure omitted; refer to PDF] determines the direction of Hopf bifurcation: the Hopf bifurcation is supercritical (subcritical) if [figure omitted; refer to PDF] and the bifurcating periodic solutions exist for [figure omitted; refer to PDF] ; [figure omitted; refer to PDF] determines the stability of the bifurcating periodic solutions: if [figure omitted; refer to PDF] , the bifurcating periodic solutions are stable (unstable); and [figure omitted; refer to PDF] determines the period of the bifurcating periodic solutions: the period increases (decrease) if [figure omitted; refer to PDF] [8-11].

4. Numerical Simulations

In this section, in order to illustrate the results above, we will give two examples.

Example 1.

In system (2), we choose that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] ; then, [figure omitted; refer to PDF] in which [figure omitted; refer to PDF] with initial and Neumann boundary conditions [figure omitted; refer to PDF]

What should be remarked is that we choose the parameter values stochastically under the condition [figure omitted; refer to PDF] in order to ensure the existence of Hopf bifurcation at [figure omitted; refer to PDF] when [figure omitted; refer to PDF] .

So, [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Then, we can know on the basis of Theorem 1 that the trivial uniform steady state [figure omitted; refer to PDF] is asymptotically stable when [figure omitted; refer to PDF] . When [figure omitted; refer to PDF] , the steady state is unstable and a Hopf bifurcation is arising from the steady state. The numerical simulations in Figures 1 and 2 illustrate the facts.

Figure 1: The temporal solution found by numerical integration of systems (64) and (66) with [figure omitted; refer to PDF] : (a) [figure omitted; refer to PDF] and (b) [figure omitted; refer to PDF] .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

Figure 2: The temporal solution found by numerical integration of systems (64) and (66) with [figure omitted; refer to PDF] : (a) [figure omitted; refer to PDF] and (b) [figure omitted; refer to PDF] .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

When [figure omitted; refer to PDF] , we get that [figure omitted; refer to PDF] ; then, we can acquire that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Hence, when [figure omitted; refer to PDF] passes through [figure omitted; refer to PDF] to the right [figure omitted; refer to PDF] , the bifurcation turns up, and the corresponding periodic orbits are orbitally asymptotically stable.

Example 2.

In system (2), we choose that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] ; then, [figure omitted; refer to PDF] in which [figure omitted; refer to PDF] with initial and Dirichlet boundary conditions [figure omitted; refer to PDF]

The similar Hopf bifurcation phenomenon is illustrated by the numerical simulations in Figures 3 and 4.

Figure 3: The temporal solution found by numerical integration of systems (67) and (69) with [figure omitted; refer to PDF] : (a) [figure omitted; refer to PDF] and (b) [figure omitted; refer to PDF] .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

Figure 4: The temporal solution found by numerical integration of systems (67) and (69) with [figure omitted; refer to PDF] : (a) [figure omitted; refer to PDF] and (b) [figure omitted; refer to PDF] .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

5. Discussion and Research Perspective

This section is devoted to a summary of discussion and research perspective for the generalized reaction-diffusion neural network model. The model is based on the assumption that the signal transmission is of a digital (McCulloch-Pitts) nature; the model then described a combination of analog and digital signal processing in the network [12]. Depending on the modeling approaches, neural networks can be modeled either as a static neural network model or as a local field neural network model. In order to let the problem be more general in many aspects, we build a generalized reaction-diffusion neural network model which includes reaction-diffusion local field neural networks and reaction-diffusion static neural networks. For a delayed neural network, an important issue is the dynamical behaviors of the network [13]. Thus, there has been a large body of work discussing the stability and bifurcation in delayed neural network models. By analyzing the characteristic equation, we discussed the local stability of the trivial uniform of system (2) [14]. It was shown that when the delay [figure omitted; refer to PDF] varies, the trivial uniform steady state exchanges its stability and Hopf bifurcations occur. Numerical simulations illustrated the occurrence of the bifurcate periodic solutions when the delay [figure omitted; refer to PDF] passes the critical value [figure omitted; refer to PDF] .

A research perspective includes the problem of determining the bifurcating periodic solutions and the stability and directions of the Hopf bifurcation using the normal form theory and the center manifold reaction. A challenging perspective is the comparison of the generalized model introduced in the present paper with the experimentally measurable quantities. Indeed, the mathematical models should reproduce both qualitatively and quantitatively empirical data (see [4]).

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 61305076) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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