Stability and Hopf Bifurcation Analysis of Coupled Optoelectronic Feedback Loops

and Applied Analysis 3 Define


Introduction
In recent research [1][2][3][4][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][7][8][9][10][11][12][13][14][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,  1 () is the normalized voltage signal applied to the electrooptic modulator,  is the feedback time delay,  1 and  2 are the filter low-pass and high-pass corner frequencies,  is the dimensionless feedback strength, they are all positive constants, and  0 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),  > 0 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 [16], 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. [17] 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.

Stability Analysis
In this section, we consider the linear stability of the nonlinear coupled system It is easy to see that (0, −cos 2  0 , 0, −cos 2  0 ) is the only equilibrium of system (3).Linearizing system (3) around  and denote  = sin 2 0 , we get the linearization system and the characteristic equation of system (4) which is equivalent to Notice that when  = 0, (5) becomes whose roots are So, we have the following lemma.
Next, we regard  as the bifurcation parameter to investigate the distribution of roots of ( 6) and (7).
Let  = i ( > 0) be a root of (6) and substituting  = i into (6), separating the real and imaginary parts yields Hence, (11) has a sequence of roots {  } ≥0 (see Figure 1), and Define Then, (  ,   ) is the solution of (10).From (10), we know that which gives that From Figure 1, we know that   → ∞ when  → ∞, which means that  2  >  1  2 ; furthermore,  is increasing with respect to , when  is sufficiently big.
Proof.Substituting () into (6) and taking the derivative with respect to , it follows that Therefore, noting that and by a straight computation, we get where As to (7), it can be easily found that − 1 , − 2 are two negative roots when  = 1, so, next, we only focus on (7) with  ̸ = 1.
Repeat the previous process, we have
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 [18], we have the following results on stability and bifurcation to system (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. [17].
Let () and  * () be eigenvectors of (0) and  * associated to  * and − * , respectively.It is not difficult with verify that where Then, ⟨ * (), ()⟩ = 1, ⟨ * (), ()⟩ = 0. Let   be the solution of (39) and define On the center manifold C 0 , we have where and  are local coordinates for center manifold C 0 in the direction of  * and  * .Note that  is real if   is real.We only consider real solutions.
For solution   in C 0 , since  = 0, we have We rewrite this equation as where Notice that where  () (, , ) =  ()  20 () Combing (38) and by straightforward computation, we can obtain the coefficients which will be used in determining the important quantities: +  *   (1)  20 (−) 2 +  − *   (1)  11 (−)) + (1 − ) 2  (2 − *   (3)  11 (−) We still need to compute  20 () and  11 (), for  ∈ [−, 0).We have Then, from (52), we get which implies that Here,  and  are both four-dimensional vectors and can be determined by setting  = 0 in (, , ).In fact, from (38) and we have It follows from (52) and the definition of  that which implies that Consequently, the above  21 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  0 .The direction and stability of the Hopf bifurcation in the center manifold can be determined by  2 and  2 , respectively.In fact, if  2 > 0 ( 2 < 0), then the bifurcating periodic solutions are forward (backward); the bifurcating periodic solutions on the center manifold are stable (unstable) if  2 < 0 ( 2 > 0); and  2 determines the period of the bifurcating periodic solutions: the period increases (decreases) if  2 > 0 ( 2 < 0).
From the discussion in Section 2, we have known that Re   (  ) > 0; therefore; we have the following result.

Theorem 7. The direction of the Hopf bifurcation for system
(3) at the equilibrium (0, −cos 2  0 , 0, −cos 2  0 ) when  =  * is forward (backward), and the bifurcating periodic solutions on the center manifold are stable (unstable) if Re( 1 (0)) < 0 (> 0).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  =  0 .

Numerical Simulations
In this section, we will carry out numerical simulations on system (3) at special values of .We choose a set of data as follows: which are the same as those in [1].Then,  = 1,  = 0.
From the analysis in Section 2, we know that () is increasing with respect to () when () >  1  2 , which means that that is,  0 ( 0 ) is the first critical value at which system (3) undergoes a Hopf bifurcation.
When  = 1.9, by the previous results, it follows that Hence, we arrive at the following conclusion: the equilibrium  is asymptotically stable when  ∈ [0, 1.1008) and unstable when  ∈ (1.1008, +∞), 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 Then, we have the following: the equilibrium  is asymptotically stable when  ∈ [0, 0.6093), and unstable when  ∈ (0.6093, +∞), 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).

Conclusion
Ravoori et al. [1] 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,  1 () and  1 () are uncoupled with equations  2 () and  2 (), 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  0 is independent of .The analysis of characteristic equation (7) shows that the value of  can affect the first critical value  0 definitely.And we draw a conclusion that when  is in an interval, in which  0 <  0 holds, solutions of system (1) and (2) keep synchronous; when  belongs to the interval, in which  0 <  0 holds, solutions of system (1) and (2) can also keep synchronous with  <  0 , while they lose their synchronization when  >  0 , no matter whether  <  0 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.