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.
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:
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:with initial and boundary conditions (Neumann boundary conditions):where , , and are random constants, where and represent the neuron charging time constants, represents the signal transmission time delay, and represent the smooth diffusion operators, , , , and represent connecting weight coefficients, and , , , and represent the coefficients of , , , and , respectively. , and are the state variables and space variable, respectively. and are the action functions of the neurons satisfying . is a bounded domain in with smooth boundary , where denotes the outward normal derivative on .
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 .
Here, we use as the eigenvalues of the operator on with the homogeneous Neumann boundary conditions and as the eigenspace corresponding to in . Let , let be an orthonormal basis of , and let . Then,
Let , , where
First, we linearize system (2) at . Then, . is invariant under the operator for each , and is an eigenvalue of if and only if it is an eigenvalue of the matrix for some , in which case, there is an eigenvalue in .
The characteristic equation of is of the formwhere
Letting , then (6) becomes
Obviously,
Obviously, if the following holds:then , . Hence, if holds, when , the trivial uniform steady state of problems (2) and (3) is locally stable.
Let be a solution of (6), separating real and imaginary parts; then, we can get that
Squaring and adding the two equations of (11), we obtain that
Letting , then (12) becomes
Obviously, it is easy to calculate that
Let
Therefore, if , (13) has no positive roots. Then, if and Holds, the trivial uniform steady state of system (2) is locally asymptotically stable for all and .
For , if , then (12) has a unique positive root , where
It means that the characteristic equation (6) admits a pair of purely imaginary roots of the form for .
Take . Obviously, (12) holds if and only if . Now, we define that
Then, for , when , (6) has a pair of purely imaginary roots and all roots of it have negative real parts for . It is easy to see that if holds, the trivial uniform steady state is locally stable for . Hence, on the basis of the general theory on characteristic equations of delay-differential equations from [3, Theorem 4.1], we can know that remains stable when , where .
Now, we claim that
This will mean that there exists at least one eigenvalue with positive real part when . 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 ; then,
So, we know that
Therefore,
By (11), we can obtain that
Because , so
Hence, the transversal condition holds and a Hopf bifurcation occurs when and .
Consequently, we gain the following results.
Theorem 1. Let and let be defined by (15). For system (2), let hold. If , the trivial uniform steady state of system (2) is locally asymptotically stable when ; if , the trivial uniform steady state is asymptotically stable for and is unstable for ; furthermore, system (2) undergoes a Hopf bifurcation at when .
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 at the critical value of . In this section, we derive explicit formulae to determine the properties of the Hopf bifurcation at critical value by using the normal form theory and center manifold reduction for PFDEs. In this section, we also let the condition hold and . And the work of Bianca and Guerrini in papers [4–7] is the founder of the method in this section.
Set . We first should normalize the delay by the time-scaling . Then, (2) can be rewritten in the fixed phase space as where is defined bywhere .
By the discussion in Section 2, we can know that the origin is a steady state of (24) and are a pair of simple purely imaginary eigenvalues of the linear equation and the functional differential equation
On the basis of the Riesz representation theorem, there exists a function of bounded variation for such that
Here, we choose thatwhere is the Dirac delta function.
Let denote the infinitesimal generator of the semigroup induced by the solutions of (27) and let be the formal adjoint of under the bilinear pairingwhere , , . Then, and are a pair of adjoint operators.
By the discussions in Section 2, we can realize that has a pair of simple purely imaginary eigenvalues and they are also eigenvalues of since and are adjoint operators. Let and be the center spaces of and associated with , respectively. Hence, is the adjoint space of and .
Letthen,is a basis of associated with and is a basis of associated with .
Let , wherefor , and let , wherefor .
Now we define that , and construct a new basis for by
Hence, , which is the second-order identity matrix. Moreover, we define for and for . Then, the center space of linear equation (26) is given by , whereand denotes the complementary subspace of , where
Let be defined bywhere is given by
Then, we have rewritten system (24), and it can be rewritten as follows:
The solution of (24) on the center manifold is given by
Letting , , thenwhere
We can use some easy computations to show thatwhere
Setting , by calculating, we get that
Because there are and in where , we still need to compute them.
By [4], we know thatwherefor , with . It follows from (43), (48), and (49) that
By (49), we have that for
Comparing the coefficients with (49), we get that for
By (50), (52), and the definition of , we get that
Noticing that , hence,where which is a constant vector.
In a similar way, by (50) and (53), we have thatwhere which is also a constant vector.
In what follows, we seek appropriate and . From the definition of and (50), we can obtain thatwhere
Substituting (55) and (59) into (57), we can obtain that
In a similar way, substituting (56) and (60) into (58), we obtain that
Therefore, we can compute the following values:which determine the quantities of bifurcating periodic solutions in the center manifold at the critical value ; that is, determines the direction of Hopf bifurcation: the Hopf bifurcation is supercritical (subcritical) if and the bifurcating periodic solutions exist for ; determines the stability of the bifurcating periodic solutions: if , the bifurcating periodic solutions are stable (unstable); and determines the period of the bifurcating periodic solutions: the period increases (decrease) if [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 , , , , , , , , , , and ; then,in which with initial and Neumann boundary conditionsWhat should be remarked is that we choose the parameter values stochastically under the condition in order to ensure the existence of Hopf bifurcation at when .
So, and . Then, we can know on the basis of Theorem 1 that the trivial uniform steady state is asymptotically stable when . When , 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.
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(b)
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When , we get that ; then, we can acquire that and . Hence, when passes through to the right , the bifurcation turns up, and the corresponding periodic orbits are orbitally asymptotically stable.
Example 2. In system (2), we choose that , , , , , , , , and ; then,in whichwith initial and Dirichlet boundary conditionsThe similar Hopf bifurcation phenomenon is illustrated by the numerical simulations in Figures 3 and 4.
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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 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 passes the critical value .
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]).
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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.