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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 367589, 9 pages
http://dx.doi.org/10.1155/2013/367589
Research Article

Bifurcation Analysis in a Two-Dimensional Neutral Differential Equation

Department of Mathematics, Northeast Forestry University, Harbin 150040, China

Received 2 February 2013; Accepted 5 April 2013

Academic Editor: Chunrui Zhang

Copyright © 2013 Ming Liu and Xiaofeng Xu. 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.

Abstract

The dynamics of a 2-dimensional neural network model in neutral form are investigated. We prove that a sequence of Hopf bifurcations occurs at the origin as the delay increases. The direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are determined by using normal form method and center manifold theory. Global existence of periodic solutions is established using a global Hopf bifurcation result of Krawcewicz et al. Finally, some numerical simulations are carried out to support the analytic results.

1. Introduction

The fundamental theories and applications of neutral functional differential equations have been largely developed and there are a great deal of important conclusions on the stability, bifurcation theory, and numerical solutions of neutral delay differential equations during the last few decades. It is well known that Wei and Ruan [1] have established a basic theory to analyze the distribution of the zeros of general transcendental equations and applied it to the neutral functional differential equations. One can use the theory to study the existence of Hopf bifurcation to a neutral delay differential equation. In [2], Wang and Wei extend the computation of the properties of Hopf bifurcation, such as the direction of bifurcation and stability of bifurcating periodic solutions, of DDE introduced by Kazarinoff et al. Based on combining the global Hopf bifurcation theory of neutral equations due to Krawcewicz et al. and the higher dimensional Bendixson’s criterion for ordinary differential equations due to Li and Muldowney, the global existence of periodic solutions of neutral differential equations have been studied by Qu et al. [3]. But research findings of bifurcation about high-dimension neutral differential equations can rarely be found.

In order to determine the dynamics of artificial neural network, the two-dimensional delay neural network without self-feedback is studied, and some significant results are obtained (see [48]). In this paper, we consider the following system: where ,  and are real numbers with and . is assumed adequately smooth, for example, , and satisfies the following condition:(H1), and there exists , such that for all ,(H2) for all , for all .

In the present paper, we provide a detailed analysis of this equation. By applying the local Hopf bifurcation theory (see [9] and also [1, 2, 10, 11]), we investigate the existence of stable periodic oscillations for (1). More specially, we prove that the equilibrium loses its stability as increases, and a sequence of Hopf bifurcations occurs at the origin. Whereafter, based on the normal form and center manifold theory due to [12, 13], by using the method introduced in Wang and Wei [2], we derive a sufficient condition for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions. Furthermore, the existence of periodic solutions for far away from the Hopf bifurcation values is also established, by using a global Hopf bifurcation result due to Krawcewicz et al. [14] (also see [15, 16]).

The rest of this paper is organized as follows: in Section 2, taking and as parameters, we give the analysis of stability and bifurcations at equilibria. Section 3 is devoted to establishing the direction and stability of Hopf bifurcation. Finally, a global Hopf bifurcation is established, and some numerical simulations are carried out to illustrate the analytic results in Section 4.

2. Stability and Bifurcation Analysis

In this section, we will investigate the stability and bifurcations of the equilibria by taking and as bifurcation parameters.

Under the given hypothesis (H1), it is easy to check that is an equilibrium point of system (1). At this trivial equilibrium, the linearization of system (1) is where And the corresponding characteristic equation is that is,

Case 1. Choose as parameter.
For convenience, we give two claims at first.

Claim 1. All the roots of (4) with have negative real parts for any .

Proof. When and , (4) becomes , and the roots are .
is a zero of with if and only if solves Separating the real and imaginary parts gives that , which contradicts with . Then the conclusion follows Wei and Ruan [1], and the proof is complete.

Claim 2. Equation (4) has at least two positive roots when .
In fact, the conclusion follows from and .
When is a root of (4) if and only if satisfies or
Applying the method in [3], we have that there exists a unique value of , denoted by (), satisfying (7) and a unique value of , denoted by (), satisfying (8) on the interval   when . We also can show that with and with .
Define make sense since , and and are all dependent on . Hence, () are a pair of purely imaginary roots of (4) with (). And we have the transversal condition
Moreover, it is fulfilled that , following the fact that . Now we can distinguish two cases. First, . From (7) and (8), we can obtain that , which implies and . Second, . With the similar process as above, it is obtained that and .
Summarizing the discussion above and applying Claims 1 and 2, we have the following.

Lemma 1. There exist sequence values of defined by (9) such that all the roots of (4) have negative real parts when , and (4) has at least two roots with positive real part when , a pair of purely imaginary roots when , where when .

Lemma 2. Equation (1) undergoes a pitchfork bifurcation at when .

Proof. Let . Then we have , , and . From (H1) and (H2), there exists a unique zero of , and (1) has only one steady-state when . On the other hand, when , we have and when . Thus, there exists a unique satisfying , and (1) has exactly nonzero equilibria , , and . This completes the proof.

In the following, we will investigate the stability of , , and .

Firstly, we consider the stability of when . (It has similar results with and .)

The characteristic equation of linearization of (1) at is

Equation (11) has at least one positive root for any . In fact, we have and from the proof of Lemma 2. Then the conclusion follows from and , which implies that is unstable.

Secondly, we consider the stability of when . (It has similar results with and ).

The characteristic equation of linearization of (1) at is In particular, which implies

Moreover, (12) has no zero root from .

In order to prove that is asymptotically stable for all , what we need is to verify that (12) has no purely imaginary roots. Let be the root of (12), then solves or This leads to The assertion follows.

Applying Lemmas 1 and 2, we have the following results.

Theorem 3. (i) The zero solution of (1) is asymptotically stable when and unstable when .
(ii) Equation (1) undergoes a pitchfork bifurcation at when . More precisely, some new equilibria bifurcate from zero and , are unstable, and are asymptotically stable for when .
(iii) Equation (1) undergoes a Hopf bifurcation at when .

Case 2. Regard as parameter.
First of all, we know that the root of (4) with satisfies that when , and when .
Let be a root of (4); then it follows that This leads to , makes sense when . DefineThen is a purely imaginary root of (4) with defined by (19).
Let be the root of (4), satisfying Differentiating both side of (4) gives that This implies that .
Summarizing the discussions above, one can obtain the following.

Lemma 4. (i) If  , then all roots of (4) have negative real parts.
(ii) If  , then there exist a sequence values of defined by (19) such that (4) has a pair of purely imaginary roots when . Additionally, if , then all roots of (4) have negative real parts when , all roots of (4), except , have negative real parts when , and (4) has at least a pair of root with positive real parts when , where when ; if , then (4) has at least two positive roots.

Spectral properties in Lemma 4 immediately lead to the dynamics near the origin described by the following theorem.

Theorem 5. For (1), the following holds.(i)If , then is asymptotically stable for all .(ii)If , then is asymptotically stable when , and unstable when .(iii)If , then is always unstable for all .(iv)If , then (1) undergoes a Hopf bifurcation at when .

3. Properties of Hopf BifurCation

Theorems 3(iii) and 5(iv) in the previous section give the sufficient conditions for (1) to undergo Hopf bifurcations with and as bifurcation parameters. In this section, we will investigate the direction of Hopf bifurcations and stability of bifurcating periodic solutions, following the same algorithms as Wang and Wei’s recent work and using a computation process similar to that in [2] (see also [17]). We should mention that we will choose as bifurcation parameter, and the similar results follow when choosing other coefficients as bifurcation parameters.

Let , and , then (1) becomes where , , and are denoted by (2) and . And the corresponding characteristic equation around is Comparing (23) with (4), if is found that for . Therefore, combining this fact with Lemma 4, one has the following.

Lemma 6. Assume .(i) If , then (23) has a pair of purely imaginary roots , where and are defined by (19).(ii) Let be the root of (23), satisfying then (iii) Equation (23) has at least a pair of roots with positive real parts when for , and it has two positive roots when and . All roots of (23) with , except , have negative real parts when .

For convenience, denote . Then we know that (22) undergoes a Hopf bifurcation at origin when . For , let By the Riesz representation theorem, there exist functions and such that In fact, we can choose

Define Then (22) can be written as Clearly, (30) is an abstract ODE on the phase space of (22), where , is continuous on , and exists}.

The adjoint operator with domain For , define a bilinear form:

It is not difficult to verify that and are the eigenvectors of and corresponding to the eigenvalues and , respectively, where and .

Now we compute the center manifold at . Define then we have Equation (35) can be written in the abbreviated form as with Noting that , we havewhere . Therefore, from (26), we have Substituting the expression of into (35) and comparing its coefficients with that of (36) give that It is well known that the coefficient of third degree term of Poincaré normal form of (35) is given by (see [12]) Consequently, when or , Hence we obtain Summarizing the above analysis, we have the following theorem.

Theorem 7. For (22), the direction of Hopf bifurcation at , is supercritical (subcritical), and the bifurcating periodic solutions are asymptotically stable (unstable) when   ().

4. Global Hopf Bifurcation Analysis

Our objective in this section is to obtain the global continuation of periodic solution bifurcating from the point , for (1) by using a global Hopf bifurcation theorem given by Krawcewicz et al. [14]. For the reader’s convenience, we copy (22), which is equivalent to (1), as follows We have known that (44) undergoes a local Hopf bifurcation at the origin when . Now we begin to show that each bifurcation branch can be continued with respect to the parameter under certain conditions. To bring out the ideas in the results of subsequent part, it is convenient to introduce the following notations:, is a -periodic solution of (44),.

Denote the connected component of in , where and are defined by Lemma 6.

Lemma 8. Equation (44) has no periodic nonconstant solution when .

Proof. Let . When , we from (44) have Therefore, there exists such that Which implies that when , we have Hence we obtain .
Let . With the similar process as above, it is obtained that there exists such that .
Then the conclusion follows from , and the proof is complete.

Lemma 9. If , then all periodic solution of (44) are uniformly bounded.

Proof. Let be a periodic solution of (44) and . Then there exist and such that together with and such that
Now we take the case as an example, and it has the same result when .
By (44), (48), and (H1), we have that On one hand, we have And on the other hand, we can obtain It follows that Similarly, one can prove that Which implies that
Let . In the same way, it can be verified that
Then and are uniformly bounded, and the proof is complete.

Lemma 10. If or , then (44) has no periodic nonconstant solution of period 1, where .

Proof. Let be a periodic solution to (44) of period 1. Then it is a periodic solution to the following system of ordinary differential equations: where denotes . Denote Lemma 9 shows that the periodic solution of (44) belong to the region . Clearly, (57) is equivalent to Then From (H2), it is easy to see that when and when . Thus, the classical Bendixson’s criterion implies that (57) has no nonconstant periodic solution in the region , and the proof is complete.

Up to now, we have prepared sufficiently to state the following global Hopf bifurcation results.

Theorem 11. If and either or holds, then (44) has at least one periodic solution for .

Proof. First note that the stationary points of (44) are nonsingular and isolate centers (see [14]) under the assumption for each . By (25), there exist , , and a smooth curve , such that for all , where is defined as (23), and Let Clearly, if and such that , then ,  , and . Moreover, if we put at , we have, from Re, that the crossing number is Using the local Hopf bifurcation theorem for NFDE [14, Theorem 5.6], we conclude that the connected component through in is nonempty, and thus is unbounded by the global Hopf bifurcation theorem given by Krawcewicz et al. [14, Theorem 5.14].
Lemma 9 implies that the projection of onto the -space is bounded. Meanwhile, the projection of onto -space is bounded below from Lemma 8.
By the definition of , we know that under the assumptions that and , which implies
Applying Lemma 10, one has if for , when and either or holds. Thus in order for to be unbounded, its projection onto the -space must be unbounded. Consequently, the projection of onto the -space include for when and either or holds. The proof is complete.

Next we carry out some numerical simulations for (1).

Let , and assume that , and . From Lemma 4, it is obtained that and . Accordingly, it is known that is asymptotically stable for and unstable for , and Hopf bifurcation at the origin occurs when by Theorem 5. From Theorem 7, the direction of the Hopf bifurcation at is supercritical, and the bifurcating periodic solutions are asymptotically stable. Furthermore, according to Theorem 11, (1) with this set of parameters has at least one periodic solution when . The corresponding numerical simulation results are shown in Figures 1, 2, and 3.

fig1
Figure 1: For system (1), the equilibrium is asymptotically stable when .
fig2
Figure 2: The system (1) undergoes a Hopf bifurcation at , and the bifurcating periodic solution is asymptotically stable when .
fig3
Figure 3: The system (1) undergoes a Hopf bifurcation at with sufficiently large .

Acknowledgment

This work is supported by the Fundamental Research Funds for the Central Universities, (no. DL11AB02).

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