Journal of Applied Mathematics

Volume 2012 (2012), Article ID 257635, 20 pages

doi:10.1155/2012/257635

Research Article

Synchronized Hopf Bifurcation Analysis in a Delay-Coupled Semiconductor Lasers System

Gang Zhu1,2 and Junjie Wei¹

¹Department of Mathematics, Harbin Institute of Technology, Heilongjiang, Harbin 150001, China

²School of Education Science, Harbin University, Heilongjiang, Harbin 150086, China

Received 30 May 2012; Revised 26 August 2012; Accepted 29 August 2012

Academic Editor: Nazim Idrisoglu Mahmudov

Copyright © 2012 Gang Zhu and Junjie Wei. 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 system of two semiconductor lasers, which are delay coupled via a passive relay within the synchronization manifold, are investigated. Depending on the coupling parameters, the system exhibits synchronized Hopf bifurcation and the stability switches as the delay varies. Employing the center manifold theorem and normal form method, an algorithm is derived for determining the Hopf bifurcation properties. Some numerical simulations are carried out to illustrate the analysis results.

1. Introduction

Systems of coupled semiconductor lasers (SLs) are receiving increasing interest, because of their practical importance in more and more complex experimental devices, so coupled system are studied by many researchers [1–3]. Moreover, they are important examples of delay-coupled oscillators in general [4, 5]. The distance between the lasers always results in a time delay in the coupling. In many situations, the time delay has been neglected. However, for semiconductor lasers, this is not always justified due to their large bandwidth and fast time scales of their dynamics. It is well known that delay effects can destabilize a system, furthermore, delay may even result in chaotic dynamics as was shown in [6–8]. On the other hand, time delay in the coupling can also be used to stabilize a chaotic system [9–12]. Synchronization phenomena are common in coupled semiconductor lasers systems. Research shows that even if several individual systems behave chaotically, in the case where the systems are identical, by proper coupling, the systems can be made to evolve toward a situation of isochronal synchronism [13–18].

In this paper, we consider the Lang-Kobayashi rate equation [19–22]:

which has already been analyzed by many authors during these years from the physics point of views. Here, and are the complex electric field amplitudes of the th system and the relay, respectively, is the excess carrier density, is the linewidth enhancement factor, is the coupling strength, is the pump current, and the time scale parameter is the ratio of the carrier and the photon lifetime, is the feedback phase, and all the parameters in system (1.1) are constants.

In this paper, for the passive relay (realized through a semitransparent mirror, which only receives, reflects, and passes part of the laser from and ), we consider the algebraic equation:

Noticing the coefficient of the relay , we found that since , and are all positive constants, so similar to [22], we choose the feedback phase for simplicity, whose results are different from those of the system with only by a constant multiple. Splitting the complex electric field and using the vector , , we consider the dynamics within the synchronization manifold (SM), that is, , then we have

and (1.1) can be written in the following form:

It is found that, under certain conditions, the equilibrium of system (1.4) is unstable when the delay varies from zero, and as passes through a critical value, the equilibrium becomes asymptotically stable, and after that when passes through another critical value, the equilibrium becomes unstable again, which means that there are stability switches as is increasing. Hence, a Hopf bifurcation occurs at the equilibrium when equals to each critical value, which means system (1.4) has periodic solutions and (1.1) exhibits synchronized periodic oscillation. Since the delay is caused by the distance between the lasers and the receiver, we know that the variety of the distance may result in amplitude death (amplitude tending to zero) or periodic oscillation in the complex electric field.

The paper is organized as follows. In Section 2, using the method presented in [23], we study the stability and the existence of Hopf bifurcation of system (1.4) by analyzing the distribution of the roots of the associated characteristic equation, which is a transcendental equation. In Section 3, we use the normal form method and the center manifold theory presented in Hassard et al. [24] to analyze the direction, stability, and period of the bifurcating periodic solution at critical values of . In Section 4, some numerical simulations are carried out to illustrate the analytical results.

2. Stability Analysis

For (1.4), it is straightforward to see that is an equilibrium. Linearizing equation (1.4) around , it follows that

whose characteristic equation is given by

which is equivalent to the following two equations:

So is always a negative root. Next, we study (2.4).

Obviously, (2.4) is equivalent to

It is easy to see that the roots of (2.5) are conjugatives of (2.6), so we study (2.5) only.

When , we get the root of (2.5) easily

so we have

Lemma 2.1. The equilibrium is unstable when .

Let be a root of (2.5) and substitute into (2.5) yields

which leads to

Then, one gets

Hence,

Consequently, for , one has

Let

be the root of (2.5) satisfying , .

We have the following conclusion.

Lemma 2.2. It holds that

Proof. Substituting into (2.5) and taking the derivative with respect to , it follows that

Therefore, noting that , we have

and by a straight computation, we get

where . Notice that when and when , this completes the proof.

As to the order of the sequence , we have

Lemma 2.3. Suppose . Then, , and there exists an integer such that

Proof. Condition implies that are well defined. So it is sufficient to verify that . From

and (2.12), we have

and if , then

which is equivalent to

if and only if

So, from the condition above, we have

This implies that .

From , , and , we have

Hence, the conclusion follows.

For convenience, we make the following assumption:

),().

From lemmas 2.1–2.3 and the fact that , then by the Hopf bifurcation theorem for functional differential equations [25], we have the following results on stability and bifurcation to system (1.4).

Theorem 2.4. For system (1.4), the following hold.

(i)If is satisfied, then is unstable for all .(ii)If is satisfied, then system (1.4) undergoes a Hopf bifurcation at when , . Particularly, there exists an integer such that is unstable when with , and asymptotically stable when .

Remark 2.5. From Lemmas 2.1–2.3, we have that all other roots, except (Res. ), of (2.5) with (resp., ) have negative real parts for when () holds.

3. The Direction and Stability of Hopf Bifurcation

In Section 2, we obtained that, under the assumption (), system (1.4) undergoes a Hopf bifurcation at some critical values of . In this section, we study the direction, stability, and the period of the bifurcating periodic solutions. The method we used is based on the normal-form method and the center manifold theory presented by Hassard et al. [24].

Transform to the origin and rescale the time by to normalize the delay so that system (1.4) can be written as the form

Clearly, the phase space is . For convenience, let , and be taken in . From the analysis in Section 2, we know that is the Hopf bifurcation value for system (3.1), and is the root of the characteristic equation associated with the linearization of system (3.1) when , where either or . For , let

where

By the Rieze representation theorem, there exists a matrix, , whose elements are of bounded variation functions such that

In fact, we can choose

Then, (3.1) is satisfied.

For , define the operator as

and as

where

Then, system (3.1) is equivalent to the following operator equation:

where , , for .

For , define

For and , define the bilinear form

where . Then, and are adjoint operators.

Let be the eigenvectors of and associated with and , respectively. It is not difficult to verify that

where

and , .

Let be the solution of (3.9), and define

On the center manifold , we have

where

and are local coordinates for center manifold in the direction of and . Note that is real if is real. We only consider real solutions.

For solution in , since , we have

We rewrite this equation as

where

By (3.9) and (3.18), we have

where

Expanding the above series and comparing the coefficients, we obtain

Noticing that

we have

where

and recalling that

we have

We can obtain the coefficients which will be used in determining the important quantities:

We still need to compute and for . We have

Comparing the coefficients about gives that

Then, from (3.22), we get

which implies that

where are both three-dimensional vectors and can be determined by setting in . In fact, from

we have

It follows from (3.22) and the definition of that

Combining the conditions above, we have

which implies

Consequently, the above can be expressed by the parameters and delay in system (3.1). Thus, we can compute the following quantities:

which determine the properties of bifurcating periodic solutions at the critical value . The direction and stability of Hopf bifurcation in the center manifold can be determined by and , respectively. In fact, if , then the bifurcating periodic solutions are forward (backward); the bifurcating periodic solutions on the center manifold are stable (unstable) if ; determines the period of the bifurcating periodic solutions: the period increases (decreases) if .

From the discussion in Section 2, we have known that

From Remark 2.5, we have the following results.

Theorem 3.1. For system (1.4), suppose that condition is satisfied. Then, when , one has

(i)the Hopf bifurcation at when is backward (forward) and the bifurcation periodic solutions are stable (unstable) if ;(ii)the Hopf bifurcation at when is forward (backward) and the bifurcation periodic solutions are stable (unstable) if .

Here and are given in of Theorem 2.4.

4. Numerical Simulations

In this section, we will carry out numerical simulation on system (1.4). Set

(A)  .

Clearly, is satisfied. We have , and

Thus, the equilibrium is stable when , and unstable when . Furthermore, we get

By the algorithm derived in Section 3, we can obtain

at , and

at , respectively. These imply that the direction of Hopf bifurcations is backward when , and forward when , respectively, and the bifurcating periodic solutions are orbitally asymptotically stable. These are shown in Figures 1, 2, and 3.

(a)

(b)

(c)

(d)

(e)

Figure 1: For system (1.4) with the data (A), the Hopf bifurcation is backward at the first critical value , and the bifurcating periodic solutions are asymptotically stable, where and the initial value is taken as .

(a)

(b)

(c)

(d)

(e)

Figure 2: For system (1.4) with the data (A), the equilibrium is asymptotically stable when , where and the initial value is taken as .

(a)

(b)

(c)

(d)

(e)

Figure 3: For system (1.4) with the data (A), the Hopf bifurcation is forward at , and the bifurcating periodic solutions are asymptotically stable, where and the initial value is taken as .

5. Conclusion

Flunkert et al. [22] explored an experimental system of two semiconductor lasers coupled via a passive relay within the synchronization manifold. They calculated the maximum transversal Lyapunov exponential and got blow-out bifurcations when the coupling strength passed through critical values.

In this paper, we also study the coupled system realized by a passive relay within the synchronization manifold. By analyzing the distribution of eigenvalues, we study the stability of the equilibrium and the existence of periodic solutions. We find that as the coupling strength increases, under the condition , the stability switch for occurs, which means that there exists a sequence values of and an integer satisfying

such that the equilibrium is asymptotically stable when , and unstable when , and the system undergoes a Hopf bifurcation at , where .

As a result, the modulation of the coupling strength and the delay (which is caused by the distance between the lasers and the relay) would be an efficient method to control the system in the complex electric field; the amplitude either vanishes or presents a periodic oscillation.

As per the coupled system which is realized by an active relay and the systems without synchronization, we will study in the future.

Acknowledgments

The authors are grateful to the anonymous referees for their helpful comments and valuable suggestions. This research is supported by the National Natural Science Foundation of China (no. 11031002).

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