Abstract
A class of two-neuron networks with resonant bilinear terms is considered. The stability of the zero equilibrium and existence of Hopf bifurcation is studied. It is shown that the zero equilibrium is locally asymptotically stable when the time delay is small enough, while change of stability of the zero equilibrium will cause a bifurcating periodic solution as the time delay passes through a sequence of critical values. Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using the normal form theory and center manifold theory. Finally, numerical simulations supporting the theoretical analysis are carried out.
1. Introduction
Based on the assumption that the elements in the network can respond to and communicate with each other instantaneously without time delays, Hopfield proposed Hopfield neural networks (HNNs) model in 1980s [1, 2]. During the past several years, the dynamical phenomena of neural networks have been extensively studied because of the widely application in various information processing, optimization problems, and so forth. In particular, the appearance of a cycle bifurcating from an equilibrium of an ordinary or a delayed neural network with a single parameter, which is known as a Hopf bifurcation, has attracted much attention (see [3–13]).
In 2008, Yang et al. [14] investigated the Bautin bifurcation of the two-neuron networks with resonant bilinear terms and without delay: where represents the state of the th neuron at time , is the connection function between two neurons, and are real parameters, and obtained a sufficient condition for a Bautin bifurcation to occur for system (1.1) by using the standard normal form theory and with Maple software. It is well known that in the implementation of networks, time delays are inevitably encountered because of the finite switching speed of signal transmission. Motivated by the viewpoint, in the following, we assume that the time delay from the first neuron to the second neuron is and back to the first neuron is , then we have the following neural networks whose delays are introduced: where represents the state of the -th neuron at time , is the connection function between two neurons, are real parameters, and are positive constants. We all know that time delays that occurred in the interaction between neurons will affect the stability of a network by creating instability, oscillation, and chaos phenomena.
The purpose of this paper is to discuss the stability and the properties of Hopf bifurcation of model (1.2). To the best of our knowledge, it is the first to deal with the stability and Hopf bifurcation of the system (1.2).
This paper is organized as follows. In Section 2, the stability of the equilibrium and the existence of Hopf bifurcation at the equilibrium are studied. In Section 3, the direction of Hopf bifurcation and the stability and periods of bifurcating periodic solutions on the center manifold are determined. In Section 4, numerical simulations are carried out to illustrate the validity of the main results. Some main conclusions are drawn in Section 5.
2. Stability of the Equilibrium and Local Hopf Bifurcations
Throughout this paper, we assume that the function satisfies the following conditions:
(H1) , and , for .
Hypothesis (H1) implies that is an equilibrium of the system (1.2) and linearized system of (1.2) takes the form The associated characteristic equation of (2.1) is where .
In the section, we consider the sum of two delays as the parameter to give some conditions that separate the first quadrant of the plane into two parts, one is the stable region another is the unstable region, and the boundary is the Hopf bifurcation curve.
In order to investigate the distribution of roots of the transcendental equation (2.2), the following Lemma that is stated in [15] is useful.
Lemma 2.1 (see [15]). For the transcendental equation as vary, the sum of orders of the zeros of in the open right half plane can change and only a zero appears on or crosses the imaginary axis.
Now we make the following assumptions:
(H2) and ;
(H3) .
Lemma 2.2. If (H1)–(H3) hold, then one has the following.
(i)When
Equation (2.2) has a simple pair of imaginary roots , where
(ii)For , all roots of (2.2) have strictly negative real parts.(iii)When , (2.2) has a pair of imaginary roots and all other roots have strictly negative real parts.
Proof. Obviously, by assumption (H2), is not the root of (2.2). When , then (2.2) becomes
It is easy to see that all roots of (2.6) have negative real parts.
is a pair of purely imaginary roots of (2.2) if and only if satisfies
Separating the real and imaginary parts, we get
It follows from (2.8) that
Thus, we obtain
It is clear that is well defined if condition (H3) holds.
Denote
Let
From (2.8), we know that (2.2) with has a pair of imaginary roots , which are simple.
According, the discussion and applying the Lemma 2.1 and Cooke and Grossman [16], we obtain the conclusion (ii) and (iii). This completes the proof.
Let be a root of (2.2) near , and . Due to functional differential equation theory, for every , there exists such that is continuously differentiable in for . Substituting into the left-hand side of (2.2) and taking derivative with respect to , we have which leads to By (2.8), we get So we have
From the above analysis, we have the following results.
Lemma 2.3. Let , then the following transversality condition: is satisfied.
From Lemma 2.3, we can obtain the following lemma.
Lemma 2.4. Assume that (H3) holds. If , then (2.2) has at least one root with strictly positive real part.
Remark 2.5. In fact, Applying the lemma in Cooke and Grossman [16] and Lemma 2.3, we can easily see that if , (2.2) has roots with positive real parts.
From Lemma 2.2–2.4, we have the following results on the local stability and Hopf bifurcation for system (1.2).
Theorem 2.6. For system (1.2), let be defined by (2.4) and assume that (H1)–(H3) hold.
(i)If , then the equilibrium point of system (1.2) is asymptotically stable.(ii)If , then the equilibrium point of system (1.2) is unstable.(iii) are Hopf bifurcation values for system (1.2).
3. Direction and Stability of the Hopf Bifurcation
In the previous section, we obtained some conditions which guarantee that the two-neuron networks with resonant bilinear terms undergo the Hopf bifurcation at some values of . In this section, we shall derived the explicit formulae determining the direction, stability, and period of these periodic solutions bifurcating from the equilibrium at this critical value of , by using techniques from normal form and center manifold theory [17], Throughout this section, we always assume that system (2.1) undergoes Hopf bifurcation at the equilibrium for and then is corresponding purely imaginary roots of the characteristic equation at the equilibrium .
For convenience, let . Then is the Hopf bifurcation value of (1.2). Thus, we shall study Hopf bifurcation of small amplitude periodic solutions of (1.2) from the equilibrium point for close to 0. Without loss of generality, we assume that and let . Since our analysis is local, where and . We can consider the fixed phase space .
For , define where We expand the nonlinear part of the system (1.2) and derive the following expression: where By the representation theorem, there is a matrix function with bounded variation components such that In fact, we can choose where is the Dirac delta function.
For , define Then (1.2) is equivalent to the abstract differential equation where .
For , define
For and , define the bilinear form where . We have the following result on the relation between the operators and .
Lemma 3.1. and are adjoint operators.
Proof. Let and It follows from (3.10) and the definitions of and that This shows that and are adjoint operators and the proof is complete.
By the discussions in the Section 2, we know that are eigenvalues of and they are also eigenvalues of corresponding to and , respectively. We have the following result.
Lemma 3.2. The vector where is the eigenvector of corresponding to the eigenvalue , and where is the eigenvector of corresponding to the eigenvalue , moreover, , where
Proof. Let be the eigenvector of corresponding to the eigenvalue and be the eigenvector of corresponding to the eigenvalue , namely, and . From the definitions of and , we have and . Thus, and . In addition, That is, Therefore, we can easily obtain and so hence On the other hand, Namely, Therefore, we can easily obtain and so hence In the sequel, we will verify that . In fact, from (3.10), we have
Next, we use the same notations as those in Hassard et al. [17] and we first compute the coordinates to describe the center manifold at . Let be the solution of (1.2) when .
Define on the center manifold , and we have where and and are local coordinates for center manifold in the direction of and . Noting that is also real if is real, we consider only real solutions. For solutions of (1.2), That is, where Hence, we have where Noticing and , we have From (3.33) and (3.34), we have and we obtain For unknown in , we still need to compute them.
Form (3.8), (3.32), we have where Comparing the coefficients, we obtain And we know that, for , Comparing the coefficients of (3.41) with (3.44) gives that From (3.42), (3.45), and the definition of , we get Noting that , we have where is a constant vector.
Similarly, from (3.43), (3.46), and the definition of , we have where is a constant vector.
In what follows, we will seek appropriate , in (3.48), (3.50), respectively. It follows from the definition of and (3.45), (3.46) that where .
From (3.42), we have where From (3.43), we have where Noting that and substituting (3.48) and (3.53) into (3.51), we have That is, then Hence, where Similarly, substituting (3.49) and (3.55) into (3.52), we have Then, That is, Hence, where From (3.48), (3.50), we can calculate and derive the following values: These formulae give a description of the Hopf bifurcation periodic solutions of (1.2) at on the center manifold. From the discussion above, we have the following result.
Theorem 3.3. The periodic solution is supercritical (subcritical) if ; The bifurcating periodic solutions are orbitally asymptotically stable with asymptotical phase (unstable) if ; the periods of bifurcating periodic solutions increase (decrease) if .
4. Numerical Examples
In this section, we present some numerical results to verify the analytical predictions obtained in the previous section. As an example, we consider the following special case of the system (1.2) with the parameters . Then, the system (1.2) becomes By some complicated computation by means of Matlab 7.0, we get ,, . We can easily obtain ,,,. Thus, we can calculate the following values: ,,,. We obtain that the conditions indicated in Theorem 2.6 are satisfied. Furthermore, it follows that and . Thus, the equilibrium is stable when as illustrated by the computer simulations (see Figures 1, 2, and 3). When passes through the critical value , the equilibrium loses its stability and a Hopf bifurcation occurs, that is, a family of periodic solutions bifurcations from the equilibrium . Since and , the direction of the Hopf bifurcation is and these bifurcating periodic solutions from at are stable, which are depicted in Figures 4, 5, and 6.
5. Conclusions
In this paper, we have analyzed a two-neuron networks with resonant bilinear terms. Firstly, we obtained the sufficient conditions to ensure local stability of the equilibrium and the existence of local Hopf bifurcation. Moreover, we note also that, if the two-neuron networks with resonant bilinear terms begin with a stable equilibrium, but then become unstable due to delay, then it will likely be destabilized by means of a Hopf bifurcation which leads to periodic solutions with small amplitudes. Finally, the direction of Hopf bifurcation and the stability of the bifurcating periodic orbits are discussed by applying normal form theory and center manifold theorem.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (no. 10771215 and no. 10771094), Scientific Research Fund of Hunan Provincial Education Department (no. 10C0560) and Doctoral Foundation of Guizhou College of Finance and Economics (2010).