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Stability and Hopf Bifurcation Analysis of Coupled Optoelectronic Feedback Loops
The dynamics of a coupled optoelectronic feedback loops are investigated. Depending on the coupling parameters and the feedback strength, the system exhibits synchronized asymptotically stable equilibrium and Hopf bifurcation. Employing the center manifold theorem and normal form method introduced by Hassard et al. (1981), we give an algorithm for determining the Hopf bifurcation properties.
In recent research [1–5], it is found 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 exact isochronal synchronism. Synchronization phenomena are common in coupled semiconductor systems, and they are important examples of oscillators in general, and many works are concerned with coupled semiconductor systems [6–15].
We consider a feedback loop comprises a semiconductor laser that serves as the optical source, a Mach-Zehnder electrooptic modulator, a photoreceiver, an electronic filter, and an amplifier. The dynamics of the feedback loop can be modeled by the delay differential equations [14, 15]: Here, is the normalized voltage signal applied to the electrooptic modulator, is the feedback time delay, and are the filter low-pass and high-pass corner frequencies, is the dimensionless feedback strength, they are all positive constants, and is the bias point of the modulator.
Depending on the value of the feedback strength and delay , the loop, which is modeled by system (1), is capable of producing dynamics ranging from periodic oscillations to high-dimensional chaos [1, 14, 15].
We couple two nominally identical optoelectronic feedback loops unidirectionally, that is, the transmitter affects the dynamics of the receiver but not vice versa. Thus, the equations of motion describing the coupled system are given by (1) for the transmitter and for the receiver. In (2), denotes the coupling strength. We will find that with the variety of , the dynamical behavior of the coupled system can be different, while the feedback strength keeps the same value.
The paper is organized as follows. In Section 2, using the method presented in , we study the stability, and the local Hopf bifurcation of the equilibrium of the coupled system (1) and (2) by analyzing the distribution of the roots of the associated characteristic equation. In Section 3, we use the normal form method and the center manifold theory introduced by Hassard et al.  to analyze the direction, stability and the period of the bifurcating periodic solutions at critical values of . In Section 4, some numerical simulations are carried out to illustrate the results obtained from the analysis. In Section 5, we come to some conclusion about the effect caused by the variety of parameters.
2. Stability Analysis
In this section, we consider the linear stability of the nonlinear coupled system
It is easy to see that is the only equilibrium of system (3). Linearizing system (3) around and denote , we get the linearization system and the characteristic equation of system (4) which is equivalent to
Notice that when , (5) becomes whose roots are So, we have the following lemma.
Lemma 1. The equilibrium is asymptotically stable when .
Define Then, is the solution of (10).
Reorder the set such that and is correspondent of . Then, we have the following lemma.
Let be the root of (6) satisfying and . We have the following conclusion.
Lemma 3. .
Proof. Substituting into (6) and taking the derivative with respect to , it follows that Therefore, noting that , we have and by a straight computation, we get where
Let be a root of (7). Using the same method above, we get
When , (23) has a sequence of roots , and
Define Then, is the solution of (22).
Repeat the previous process, we have
Reorder the set such that and is correspondent of .
Lemma 5. .
Compare , and reorder the set and and remove the “−” of , such that then from previous lemmas and the Hopf bifurcation theorem for functional differential equations , we have the following results on stability and bifurcation to system (3).
3. The Direction and Stability of the Hopf Bifurcation
In Section 2 we obtained some conditions under which system (3) undergoes the 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 introduced by Hassard et al. .
Move to the origin and denote , , then system (3) can be written as the form
Clearly, the phase space is . For convenience, let and , . From the analysis above we know that is the Hopf bifurcation value for system(30). Let be the root of the characteristic equation associate with the linearization of system (30) when . 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, (30) is satisfied.
For , define the operator as and as where Then, system (30) is equivalent to the following operator equation: where , , for .
For , define
For and , define the bilinear form where . Then, and are adjoint operators.
Let and be eigenvectors of and associated to and , respectively. It is not difficult with verify that where Then, , .
Let be the solution of (39) 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.
where Combing (38) and by straightforward computation, 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 (52), we get which implies that Here, and are both four-dimensional vectors and can be determined by setting in . In fact, from (38) and we have It follows from (52) and the definition of that which implies that Consequently, the above can be expressed by the parameters and delay in system (30). Thus, we can compute the following quantities: which determine the properties of bifurcating periodic solutions at the critical value . The direction and stability of the 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 ; and determines the period of the bifurcating periodic solutions: the period increases (decreases) if .
From the discussion in Section 2, we have known that ; therefore; we have the following result.
Theorem 7. The direction of the Hopf bifurcation for system (3) at the equilibrium when is forward (backward), and the bifurcating periodic solutions on the center manifold are stable (unstable) if . Particularly, the stability of the bifurcation periodic solutions of system (3) and the reduced equations on the center manifold are coincident at the first bifurcation value .
4. Numerical Simulations
Then, we can obtain From the analysis in Section 2, we know that is increasing with respect to when , which means that that is, is the first critical value at which system (3) undergoes a Hopf bifurcation.
When , by the previous results, it follows that
Hence, we arrive at the following conclusion: the equilibrium is asymptotically stable when and unstable when , and, at the first critical value, the bifurcating periodic solutions are asymptotically stable, and the direction of the bifurcation is forward (see Figures 2 and 3).
When , we can get
Then, we have the following: the equilibrium is asymptotically stable when , and unstable when , and, at the first critical value, the bifurcating periodic solutions are asymptotically stable, and the direction of the bifurcation is forward (see Figures 4, 5, and 6).
Ravoori et al.  explored an experimental system of two nominally identical optoelectronic feedback loops coupled unidirectionally, which are described by system (3). In the experiment, they found that depending on the value of the feedback strength and delay , system (1) is capable of producing dynamics ranging from periodic oscillations to high-dimensional chaos [14, 15].
This paper investigates the stability and the existence of periodic solutions. We find that with the variety of the coupling strength , even if all other parameters keep the same, the dynamical behavior can change greatly. In fact, it is clear that the first two equations, and are uncoupled with equations and , so system (1) are independent of (2), which means that coupling strength does not appear in (1). The characteristic equation of (1) has the same form as (6), so the first critical value is independent of . The analysis of characteristic equation (7) shows that the value of can affect the first critical value definitely. And we draw a conclusion that when is in an interval, in which holds, solutions of system (1) and (2) keep synchronous; when belongs to the interval, in which holds, solutions of system (1) and (2) can also keep synchronous with , while they lose their synchronization when , no matter whether or not.
As a result, the modulation of the coupling strengths together with the feedback strength would be an efficient and an easily implementable method to control the behavior of the coupled chaotic oscillators.
This paper is supported by the National Natural Science Foundation of China (no. 11031002) and the Research Fund for the Doctoral Program of Higher Education of China (no. 20122302110044).
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Copyright © 2013 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.