Abstract
A Cohen-Grossberg neural network with discrete delays is investigated in this paper. Sufficient conditions for the existence of local Hopf bifurcation are obtained by analyzing the distribution of roots of characteristic equation. Moreover, the direction and stability of Hopf bifurcation are obtained by applying the normal form theory and the center manifold theorem. Numerical simulations are given to illustrate the obtained results.
1. Introduction
In recent years, more and more mathematicians, biologists, physicists, and computer scientists focus on artificial neural networks. It is well known that the analysis of the dynamical behaviors is a necessary step for practical design of neural networks since their applications heavily depend on the dynamical behaviors; many important results on dynamical behaviors of neural networks have been obtained [1–23]. The neural networks are large-scale and complex systems, and the dynamical behaviors of neural networks with delays are more complicated; in order to obtain a deep and clear understanding of the dynamics of complicated neural networks with time delays, researchers have focused on the studying of simple systems [12–22]. This is indeed very useful since the complexity found may be carried over to large neural networks.
The research on dynamical behaviors of neural networks involves not only the dynamic analysis of equilibrium but also that of periodic solution, bifurcation, and chaos; especially, the periodic oscillatory behavior of the neural networks is of great interest in many applications [2, 3]. Since periodic oscillatory can arise through the Hopf bifurcation in different system with or without time delays, it is very important to discuss the Hopf bifurcation of neural networks.
In 1983, Cohen-Grossberg [1] proposed a kind of neural networks, which are now called Cohen-Grossberg neural networks. The networks have been successfully applied to signal processing, pattern recognition, optimization, and associative memories. Recently, some results on the existence and globally asymptotical stability of periodic Cohen-Grossberg neural networks have been obtained [7–15]. However, up to now, to the best of the author’s knowledge, bifurcation of Hopfield neural networks has been discussed by many researchers [12–19], but few results on the bifurcation of Cohen-Grossberg neural networks have been obtained. Zhao discussed the bifurcation of a two-neuron discrete-time Cohen-Grossberg neural network in [20] and the bifurcation of a two-neuron continuous-time Cohen-Grossberg neural network with distributed delays in which kernel function is in [21]. We discussed the bifurcation of a two-neuron Cohen-Grossberg neural network with discrete delays in [22]. The objective of this paper is to study the following -neuron continuous-time Cohen-Grossberg neural network with discrete delays and ring architecture: where denote the state variable of the th neuron; represent amplification functions which are positive for ; denote the signal functions of the th neuron; are appropriately behaved functions; are connection weights of the neural networks; discrete delays correspond to the finite speed of the axonal signal transmission: , .
Ring architectures have been found in variety of neural structures, and they are investigated to gain insight into the mechanisms underlying the behaviors of recurrent neural networks [23].
The rest of this paper is organized as follows. Stability property and existence of Hopf bifurcation for system (1) are obtained in Section 2. Based on the normal form method and the center manifold, the formulas for the direction of Hopf bifurcation and stability of the bifurcating periodic solutions are derived in Section 3. An example is given in Section 4 to illustrate the main results, and conclusions are drawn in Section 5.
2. Stability Analysis and Existence of Local Bifurcation
Lemma 1 (see [11]). Consider the exponential polynomial where and are constants. Then as vary, the sum of the order of zeros of on the open right half plane can change only if a zero appears on or across the imaginary axis.
In the following discussion, for convenience, we denote
Throughout this paper, we assume that(H1), , and ;(H2)there exist constants, such thatfor.
From assumption (H1)-(H2) that the origin is an equilibrium of system (1).
Let
System (1) can be transformed into the following equivalent system: where .
The linear system of system (5) around the equilibrium is given by
The associated characteristic equation of system (5) is
Suppose that is a root of the characteristic equation, where is imaginary unit which satisfies . Substituting into (7), then we have
Separating the real and imaginary parts of (8), we have where in which denotes the inverse of the cotangent function.
Since for and
Hence is an increasing bijective function.
We know from the second equation in (9) that . Denote , ; then we have from the first equation in (9) that
When .
Furthermore, from the value given above, we have
Obviously, if is a root of equation (7), is also the root of equation (7). This implies that is a pair of purely imaginary roots of equation (7). On the other hand, we have from (7) that where
Hence we have
Since the roots of the characteristic equation (7) are , when , so the equilibrium of system (5) is asymptotically stable. As the parameter varies on the open right half plane can change only if a zero appears on or across the imaginary axis. According to Lemma 1 and (14), we obtain that the equilibrium of system (5) is asymptotically stable if and only if .
When , the characteristic equation of system (5) has a simple root , and all the other roots have negative real parts. A pitchfork bifurcation may occur at the origin in system (5) [16].
When , , the characteristic equation of system (3) has a pair of purely imaginary roots , and all the other roots have negative real parts. Note that due to according to (11), so, ; that is, . We also know from (17) that . System (3)undergoes a Hopf bifurcation which occurs at the origin when .
From the above discusses, Lemma 1, and the Hopf bifurcation theorem in [24] for functional differential equations, we have the following results.
Theorem 2. Under assumptions (H1)-(H2), we have the following:(1)if , the equilibrium of system (1) is asymptotically stable;(2)if , the equilibrium of system (1) is unstable;(3)if , a Hopf bifurcation occurs at the origin in system (1),where in which and satisfies the equation , .
3. Direction and Stability of Hopf Bifurcation for the Network
In this section, we will derive explicit formulas for determining the properties of the Hopf bifurcation at critical by using the normal form theory and the center manifold theorem [25], and we always make , vary with a parameter and the other ones are fixed.
We still discuss system (5). For the sake of generality, let . Denote and , dropping the bars for simplification of notation; then system (5) can be written as functional differential equation in as where , , and are given, respectively, by where where in which
From the discussions in Section 2, we know that if , system (19) undergoes a Hopf bifurcation at the equilibrium , and the associated characteristic equation of system (19) has a pair simple imaginary roots .
By the Riesz representation theorem, there exists a bounded variation function for such that
In fact, we can choose where is the Dirac delta function and
For , define
The system (19) can be transformed into the following operator equation form: where for .
Denote
For , define
For and , we define a bilinear form
Then and are adjoint operators. We know that are eigenvalues of , so are also eigenvalues of .
Now we compute the eigenvectors of and corresponding to and .
Suppose that is the eigenvector of corresponding to ; then . From (25) and (26), we know
Similarly, we know that with and is the eigenvector of corresponding to , where
Moreover, and .
Using the same notations as Hassard et al. [25], we construct the coordinates to describe the center manifold at .
Define
On the center manifold , we have where
and are local coordinates for the center manifoldin the direction of and . Note that is real if is real. We only consider real solutions.
For solution of (19), since , we have
We rewrite this as with
From (29), (36), and (39), we have where
Since where in which , .
Denote the th element of by and the th element of by
Then if follows that
Substitute (46) into (43) and comparing the coefficients in (40) with those in (43), we have where is th element of shown in (32).
Since there are and in , we still need to figure them out. Note that on the center manifold , we have
We have from (41), (42), and (48) that
Equations (39) and (41) mean for .
Comparing the coefficients with (42), we have
From (26), (49), and (50), we can obtain
So and similarly where , , , , , .
In the following, we focus on the computation of and . From (26) and (49), we know in which
We know from (41) and (43) that in which in which
Substituting (53) and (57) into (55), we have
Solving this we can obtain .
Similarly, we can obtain from
Based on the above analysis, we can see that each in (47) is determined by the parameters and delays for (1). Thus, we can compute the following quantities:
It is known that determines the direction of the Hopf bifurcation and determines the stability of the bifurcating periodic solutions. Since , we know if ; then the Hopf Bifurcation is supercritical (subcritical), the bifurcating periodic solutions exist for , and the bifurcating periodic solutions are stable (unstable). determines the period of the bifurcating periodic solutions: the period increases (decreases) if .
Under some conditions, the equilibrium of system (1) is globally asymptotically stable. The following result can be directly obtained from Corollary 2 in [5].
Theorem 3. Under assumptions (H1)–(H3), the equilibrium of system (1) is globally asymptotically stable if the following conditions hold.(H3)There exist constants such that for .(H4)There exist positive constants such that for .(H5)The following matrix is an -matrix:
Note that the conditions in Theorem 3 have more restrictions than those in Theorem 2. Since is an M-matrix, we have [26]; that is, ; it yields , which, together with conditions (H3) and (H4), implies that ; moreover, due to . Hence, conditions (H3)–(H5) imply that the condition in Theorem 2 holds.
4. A Numerical Example
Example 1. Consider the following Cohen-Grossberg neural network with discrete delays:
We can obtain that and furthermore we obtain that in view of bisection method by using MATLAB. It is easy to know. We also know from (3) that .
According to Theorem 2, the zero solution of system (64) is asymptotically stable when , and when , the Hopf bifurcation occurs at the origin.
Case 1. Let . ; then the zero solution of system (64) is asymptotically stable. Figure 1 shows the dynamic behaviors of system (64) with initial condition .
Case 2. Let , and . We know from Theorem 2 that the Hopf bifurcation occur at the origin; furthermore, we can obtain , so the bifurcating periodic solutions are supercritical and asymptotically stable. Figures 2 and 3 show the dynamic behaviors of system (64) with initial conditions and , respectively.
The presented numerical simulations illustrate the theoretical results.
5. Conclusions
An -neuron Cohen-Grossberg neural network with discrete delays and ring architecture is analyzed in this paper. By using as a bifurcation parameter, we show that this system undergoes a Hopf bifurcations at a critical parameter: where and satisfies the equation , . The direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are investigated by applying the normal form theory and the center manifold theorem for continuous time system. The phenomena of bifurcating periodic solutions for Cohen-Grossberg neural networks coincide with the fact that learning usually requires repetition [2], and periodic sequences of neural impulse are also of fundamental significance for the control of dynamic functions of the body such as heart beat which occurs with great regularity and breathing [19]. In this paper, we extend the results about the existence of local Hopf bifurcation in [22] to the case of a discrete-time -neuron Cohen-Grossberg system with discrete delays. In the future, the problem for the existence of global Hopf bifurcation will be expected to be solved.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The author is greatly indebted to the reviewers and the editors for their very valuable suggestions and comments which improved the quality of the presentation. This research was supported by the Hebei Provincial Natural Science Foundation of China under Grant no. A2012205028 and the Innovation Foundation of Shijiazhuang Mechanical Engineering College under Grant no. Yscx1201.