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

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]: 𝐸̇_𝑗=12(1+𝑖𝛼)𝑛_𝑗𝐸_𝑗+𝑘𝑒^𝑖𝜑𝐸_𝑌𝜏𝑡−2,𝑇𝑛̇_𝑗=𝑝−𝑛_𝑗−1+𝑛_𝑗𝐸_𝑗²,𝑗=1,2, (1.1) which has already been analyzed by many authors during these years from the physics point of views. Here, 𝐸_𝑗 and 𝐸_𝑌 are the complex electric field amplitudes of the 𝑗th system and the relay, respectively, 𝑛_𝑗 is the excess carrier density, 𝛼 is the linewidth enhancement factor, 𝑘 is the coupling strength, 𝑝 is the pump current, and the time scale parameter 𝑇=𝜏_𝑐/𝜏_𝑝 is the ratio of the carrier and the photon lifetime, 𝜑 is the feedback phase, and all the parameters in system (1.1) are constants.

In this paper, for the passive relay 𝐸_𝑌 (realized through a semitransparent mirror, which only receives, reflects, and passes part of the laser from 𝐸₁ and 𝐸₂), we consider the algebraic equation: 𝐸_𝑌(𝑡)=12𝐸₁𝜏𝑡−2+𝐸₂𝜏𝑡−2. (1.2)

Noticing the coefficient of the relay 𝐸_𝑌, we found that since 𝑘,𝑒,𝑖, and 𝜑 are all positive constants, so similar to [22], we choose the feedback phase 𝜑=0 for simplicity, whose results are different from those of the system with 𝜑≠0 only by a constant multiple. Splitting the complex electric field 𝐸_𝑗=𝑥_𝑗+𝑖𝑦_𝑗 and using the vector 𝑋_𝑗=(𝑥_𝑗,𝑦_𝑗,𝑛_𝑗), 𝑗=1,2, we consider the dynamics within the synchronization manifold (SM), that is, 𝑋₁(𝑡)=𝑋₂(𝑡)=𝑋(𝑡), then we have 𝐸_𝑌𝜏𝑡−2=𝐸(𝑡−𝜏), (1.3) and (1.1) can be written in the following form: 𝑥̇(𝑡)=𝑛(𝑡)2𝑥(𝑡)(𝑡(),𝑦̇(𝑡)=𝑛(𝑡)−𝛼𝑦)+𝑘𝑥𝑡−𝜏2(𝑡)(𝑡(),𝑇𝑛̇(𝑡)(𝑡)−[(𝑡𝑥𝛼𝑥+𝑦)+𝑘𝑦𝑡−𝜏=𝑝−𝑛1+𝑛)]²(𝑡)+𝑦²(𝑡). (1.4)

It is found that, under certain conditions, the equilibrium of system (1.4) is unstable when the delay 𝜏 varies from zero, and as 𝜏 passes through a critical value, the equilibrium becomes asymptotically stable, and after that when 𝜏 passes through another critical value, the equilibrium becomes unstable again, which means that there are stability switches as 𝜏 is increasing. Hence, a Hopf bifurcation occurs at the equilibrium when 𝜏 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 𝜏. 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 𝐸(0,0,𝑝) is an equilibrium. Linearizing equation (1.4) around (0,0,𝑝), it follows that 𝑥̇(𝑡)=𝑝2𝑥(𝑡)−𝛼𝑝2𝑦(𝑡)(),𝑦̇(𝑡)=+𝑘𝑥𝑡−𝜏𝛼𝑝2𝑥(𝑡)+𝑝2𝑦(𝑡)(),𝑛̇(𝑡)𝑛(𝑡)+𝑘𝑦𝑡−𝜏=−𝑇, (2.1) whose characteristic equation is given by 1𝜆+𝑇𝑝𝜆−2−𝑘𝑒^−𝜆𝜏2+𝛼²𝑝²4=0, (2.2) which is equivalent to the following two equations: 1𝜆+𝑇𝑝=0,𝜆−2−𝑘𝑒^−𝜆𝜏2+𝛼²𝑝²4=0. (2.3)(2.4) So 𝜆=−(1/𝑇) is always a negative root. Next, we study (2.4).

Obviously, (2.4) is equivalent to 𝑝𝜆−2−𝑘𝑒^−𝜆𝜏=𝑖𝛼𝑝2,𝑝𝜆−2−𝑘𝑒^−𝜆𝜏=−𝑖𝛼𝑝2. (2.5)(2.6)

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

When 𝜏=0, we get the root of (2.5) easily 𝑝𝜆=2+𝑘+𝑖𝛼𝑝2, (2.7) so we have

Lemma 2.1. The equilibrium 𝐸(0,0,𝑝) is unstable when 𝜏=0.

Let 𝜆=𝑖𝜔(𝜔≠0) be a root of (2.5) and substitute 𝜆=𝑖𝜔 into (2.5) yields 𝑝2+𝑘cos𝜔𝜏=0,𝜔+𝑘sin𝜔𝜏=𝛼𝑝2, (2.8) which leads to 𝜔−𝛼𝑝2²=𝑘²−𝑝²4. (2.9) Then, one gets 𝜔_±=12𝛼𝑝±4𝑘²−𝑝². (2.10) Hence, cos𝜔₊𝜏=−𝑝2𝑘,sin𝜔₊𝜏=𝛼𝑝−2𝜔₊2𝑘=−4𝑘²−𝑝²,2𝑘cos𝜔₋𝜏=−𝑝2𝑘,sin𝜔₋𝜏=𝛼𝑝−2𝜔₋=2𝑘4𝑘²−𝑝².2𝑘 (2.11) Consequently, for 𝑘>(𝑝/2), one has 𝜏⁻_𝑗=1𝜔₋𝜋−arcsin4𝑘²−𝑝²,𝜏2𝑘+2𝑗𝜋_+𝑗=1𝜔₊𝜋+arcsin4𝑘²−𝑝²2𝑘+2𝑗𝜋,𝑗=0,1,2,…. (2.12) Let 𝜆(𝜏)=𝛾(𝜏)+𝑖𝜔(𝜏) (2.13) be the root of (2.5) satisfying 𝛾(𝜏^±_𝑗)=0, 𝜔(𝜏^±_𝑗)=𝜔_±.

We have the following conclusion.

Lemma 2.2. It holds that (𝑖)𝛾^′𝜏⁺_𝑗()𝛾>0𝑖𝑖^′𝜏⁻_𝑗𝑝<0,𝑤ℎ𝑒𝑛2√<𝑘<1+𝛼²2𝑝√>0,𝑤ℎ𝑒𝑛𝑘>1+𝛼²2𝑝. (2.14)

Proof. Substituting 𝜆(𝜏) into (2.5) and taking the derivative with respect to 𝜏, it follows that d𝜆d𝜏+𝜆𝑘𝑒^−𝜆𝜏+𝑘𝜏𝑒^−𝜆𝜏d𝜆d𝜏=0. (2.15) Therefore, noting that 𝑘𝑒^−𝜆𝜏=𝜆−(𝑝/2)−(𝑖/2)𝛼𝑝, we have d𝜆=−()d𝜏−𝜆𝜆−𝑝/2𝑖/2𝛼𝑝1+𝜏𝜆−𝑝/2−(𝑖/2)𝛼𝑝, (2.16) and by a straight computation, we get 𝛾^′𝜏^±_𝑗=𝜔_±Δ𝜔_±−𝛼𝑝2𝜔=±^±Δ𝑘²−𝑝²4, (2.17) where Δ=(1+(𝑝𝜏/2))²+𝜏²(𝜔−(𝛼𝑝/2))². Notice that 𝜔₋>0 when (𝑝/2)<𝑘<(√1+𝛼²/2)𝑝 and 𝜔₋<0 when 𝑘>(√1+𝛼²/2)𝑝, this completes the proof.

As to the order of the sequence {𝜏^±_𝑗}, we have

Lemma 2.3. Suppose (𝑝/2)<𝑘<(𝛼𝑝/2𝜋). Then, 𝜏⁻₀<𝜏⁺₀, and there exists an integer 𝑚≥0 such that 𝜏⁻₀<𝜏⁺₀<𝜏⁻₁<⋯<𝜏⁻_𝑚<𝜏⁺_𝑚<𝜏⁺_𝑚+1<𝜏⁻_𝑚+1. (2.18)

Proof. Condition (𝑝/2)<𝑘<(𝛼𝑝/2𝜋) implies that 𝜏^±_𝑗 are well defined. So it is sufficient to verify that 𝜏⁻₀<𝜏⁺₀. From arcsin𝑥=𝑥+^∞_𝑛=1(2𝑛−1)!!()𝑥2𝑛+1)(2𝑛!!^2𝑛+1=∶𝑥+𝐴,𝑥∈[−1,1] (2.19) and (2.12), we have 𝜏⁻₀=1𝜔₋𝜋−4𝑘²−𝑝²,𝜏2𝑘−𝐴₊₀=1𝜔₊𝜋+4𝑘²−𝑝²,2𝑘+𝐴 (2.20) and if 𝜏⁻₀<𝜏⁺₀, then 𝜔₊𝜋−4𝑘²−𝑝²2𝑘−𝐴<𝜔₋𝜋+4𝑘²−𝑝²2𝑘+𝐴, (2.21) which is equivalent to 𝛼𝑝+4𝑘²−𝑝²𝜋−4𝑘²−𝑝²<2𝑘−𝐴𝛼𝑝−4𝑘²−𝑝²𝜋+4𝑘²−𝑝²2𝑘+𝐴, (2.22) if and only if 𝜋4𝑘²−𝑝²<𝛼𝑝4𝑘²−𝑝²2𝑘+𝐴. (2.23) So, from the condition above, we have 𝑘<𝛼𝑝2𝜋⇒𝜋4𝑘²−𝑝²<𝛼𝑝4𝑘²−𝑝²2𝑘<𝛼𝑝4𝑘²−𝑝²2𝑘+𝐴. (2.24) This implies that 𝜏⁻₀<𝜏⁺₀.
From 𝜏⁺_𝑗+1−𝜏⁺_𝑗=(2𝜋/𝜔₊), 𝜏⁻_𝑗+1−𝜏⁻_𝑗=(2𝜋/𝜔₋), and 𝜔₊>𝜔₋, we have 𝜏⁺_𝑗+1−𝜏⁺_𝑗<𝜏⁻_𝑗+1−𝜏⁻_𝑗. (2.25) Hence, the conclusion follows.

For convenience, we make the following assumption:(H₁)𝑘∈(0,(𝑝/2))∪[(√1+𝛼²/2)𝑝,∞),(H₂)(𝑝/2)<𝑘<(𝛼𝑝/2𝜋).

From lemmas 2.1–2.3 and the fact that (𝛼𝑝/2𝜋)<(√1+𝛼²/2)𝑝, 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 (H₁) is satisfied, then 𝐸 is unstable for all 𝜏≥0.(ii)If (H₂) is satisfied, then system (1.4) undergoes a Hopf bifurcation at 𝐸 when 𝜏=𝜏^±_𝑗, 𝑗=0,1,…. Particularly, there exists an integer 𝑚≥0 such that 𝐸 is unstable when 𝜏∈{0}∪(∪^𝑚_𝑗=0(𝜏⁺_𝑗−1,𝜏⁻_𝑗))∪(𝜏⁺_𝑚,∞) with 𝜏⁺₋₁=0, and asymptotically stable when 𝜏∈∪^𝑚_𝑗=0(𝜏⁻_𝑗,𝜏⁺_𝑗).

Remark 2.5. From Lemmas 2.1–2.3, we have that all other roots, except 𝑖𝜔₋(Res. 𝑖𝜔₊), of (2.5) with 𝜏=𝜏⁻_𝑗(resp., 𝜏=𝜏⁺_𝑗) have negative real parts for 𝑗=0,1,…,𝑚 when (H₂) holds.

3. The Direction and Stability of Hopf Bifurcation

In Section 2, we obtained that, under the assumption (H₂), system (1.4) undergoes a Hopf bifurcation at some critical values of 𝜏. 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 𝐸(0,0,𝑝) to the origin 𝑂(0,0,0) and rescale the time by 𝑡→(𝑡/𝜏) to normalize the delay so that system (1.4) can be written as the form 𝑥̇(𝑡)𝑝=𝜏2𝑥(𝑡)(𝑡()+𝑛(𝑡)−𝛼𝑦)+𝑘𝑥𝑡−12𝑥(𝑡)(𝑡,𝑦̇(𝑡)𝑝−𝛼𝑦)=𝜏2(𝑡)(𝑡()+𝑛(𝑡)𝛼𝑥+𝑦)+𝑘𝑦𝑡−12(𝑡)(𝑡,𝑛̇(𝑡)=𝜏𝛼𝑥+𝑦)𝑇−𝑛(𝑡)−1+𝑝+𝑛(𝑡)𝑥²(𝑡)+𝑦²(𝑡). (3.1)

Clearly, the phase space is 𝒞=𝒞([−1,0],ℝ³). For convenience, let 𝜏=𝜏₀+𝜇, 𝜇∈ℝ and 𝜏₀ be taken in {𝜏⁺_𝑗}∪{𝜏⁻_𝑗}. From the analysis in Section 2, we know that 𝜇=0 is the Hopf bifurcation value for system (3.1), and 𝑖𝜔₀𝜏₀ is the root of the characteristic equation associated with the linearization of system (3.1) when 𝜏=𝜏₀, where either 𝜔₀=𝜔₊ or 𝜔₀=𝜔₋. For 𝜙=(𝜙₁,𝜙₂,𝜙₃)∈𝒞, let 𝐿_𝜇𝜙=𝜏₀+𝜇𝐵𝜙(0)+𝜏₀+𝜇𝐶𝜙(−1), (3.2) where 𝑝𝐵=2−𝛼𝑝20𝛼𝑝2𝑝20100−𝑇,𝐶=𝑘000𝑘0000. (3.3) By the Rieze representation theorem, there exists a 3×3 matrix, 𝜂(𝜃,𝜇)(−1≤𝜃≤0), whose elements are of bounded variation functions such that 𝐿_𝜇𝜙=⁰₋₁d𝜂𝜃,𝜇𝜙(𝜃),𝜙∈𝒞. (3.4) In fact, we can choose 𝜂=𝜏𝜃,𝜇⁰(),−𝜏+𝜇𝐵,𝜃=0,0,𝜃∈−1,0₀+𝜇𝐶,𝜃=−1. (3.5) Then, (3.1) is satisfied.

For 𝜙∈𝒞, define the operator 𝐴(𝜇) as 𝐴𝜇𝜙(𝜃)=d𝜙(𝜃)[),∫d𝜃,𝜃∈−1,0₀₋₁d𝜂𝑡,𝜇𝜙(𝑡),𝜃=0, (3.6) and 𝑅(𝜇)𝜙 as 𝑅𝜇𝜙(𝜃)=[),𝑓0,𝜃∈−1,0𝜇,𝜙,𝜃=0, (3.7) where 𝑓𝜇,𝜙=𝜏₀𝜙+𝜇¹(0)𝜙₃(0)2−𝛼𝜙₂(0)𝜙₃(0)2𝛼𝜙₁(0)𝜙₃(0)2+𝜙₂(0)𝜙₃(0)2−1𝑇1+𝑝+𝜙₃(0)𝜙²₁(0)+𝜙²₂(0). (3.8) Then, system (3.1) is equivalent to the following operator equation: 𝑢̇_𝑡=𝐴𝜇𝑢_𝑡+𝑅𝜇𝑢_𝑡, (3.9) where 𝑢(𝑡)=(𝑥(𝑡),𝑦(𝑡),𝑛(𝑡))^𝑇, 𝑢_𝑡=𝑢(𝑡+𝜃), for 𝜃∈[−1,0].

For 𝜓∈𝒞¹([0,1],ℝ³), define 𝐴^∗𝜓(𝑠)=−d𝜓(𝑠)(],∫d𝑠,𝑠∈0,1₀₋₁𝜓(−𝜉)d𝜂(𝜉,0),𝑠=0. (3.10)

For 𝜙∈𝒞[−1,0] and 𝜓∈𝒞[0,1], define the bilinear form 𝜓(𝑠),𝜙(𝜃)=𝜓(0)𝜙(0)−⁰₋₁^𝜃₀𝜓(𝜉−𝜃)d𝜂(𝜃)𝜙(𝜉)d𝜉, (3.11) where 𝜂(𝜃)=𝜂(𝜃,0). Then, 𝐴(0) and 𝐴^∗ are adjoint operators.

Let 𝑞(𝜃),𝑞^∗(𝑠) be the eigenvectors of 𝐴(0) and 𝐴^∗ associated with 𝑖𝜔₀𝜏₀ and −𝑖𝜔₀𝜏₀, respectively. It is not difficult to verify that 𝑞(𝜃)=1,𝛽,0^𝑇𝑒^{𝑖𝜔0𝜏0𝜃},𝑞^∗(𝑠)=1𝐷(1,𝜈,0)𝑒^{𝑖𝜔0𝜏0𝑠}, (3.12) where 𝛽=−2𝛼𝑝𝑖𝜔₀−𝑝2−𝑘𝑒^{−𝑖𝜔0𝜏0},𝜈=−2𝛼𝑝𝑖𝜔₀+𝑝2+𝑘𝑒^{𝑖𝜔0𝜏0},𝐷=1+𝛽𝜈1+𝑘𝜏₀𝑒^{−𝑖𝜔0𝜏0}, (3.13) and ⟨𝑞^∗(𝑠),𝑞(𝜃)⟩=1, ⟨𝑞^∗(𝑠),𝑞(𝜃)⟩=0.

Let 𝑢_𝑡 be the solution of (3.9), and define 𝑧(𝑡)=𝑞^∗,𝑢_𝑡,𝑊(𝑡,𝜃)=𝑢_𝑡(𝜃)−2Re𝑧(𝑡)𝑞(𝜃). (3.14) On the center manifold ℭ₀, we have 𝑊(𝑡,𝜃)=𝑊(𝑧(𝑡),𝑧(𝑡),𝜃), (3.15) where 𝑊(𝑧,𝑧,𝜃)=𝑊₂₀𝑧²2+𝑊₁₁𝑧𝑧+𝑊₀₂𝑧²2+⋯, (3.16)𝑧 and 𝑧 are local coordinates for center manifold ℭ₀ in the direction of 𝑞^∗ and 𝑞^∗. Note that 𝑊 is real if 𝑢_𝑡 is real. We only consider real solutions.

For solution 𝑢_𝑡 in ℭ₀, since 𝜇=0, we have 𝑧̇(𝑡)=𝑖𝜔₀𝜏₀𝑧+𝑞^∗(𝜃),𝑓0,𝑊+2Re𝑧(𝑡)𝑞(𝜃)=𝑖𝜔₀𝜏₀𝑧+𝑞^∗(0),𝑓0,𝑊(𝑧,𝑧,0)+2Re𝑧(𝑡)𝑞(0)=𝑖𝜔₀𝜏₀𝑧+𝑞^∗(0)𝑓₀(𝑧,𝑧). (3.17) We rewrite this equation as 𝑧̇(𝑡)=𝑖𝜔₀𝜏₀𝑧+𝑔(𝑧,𝑧), (3.18) where 𝑔(𝑧,𝑧)=𝑔₂₀𝑧²2+𝑔₁₁𝑧𝑧+𝑔₀₂𝑧²2+𝑔₂₁𝑧²𝑧2⋯. (3.19)

By (3.9) and (3.18), we have 𝑊̇=𝑢̇_𝑡−𝑧𝑞̇−𝑧̇𝑞=𝐴𝑊−2Re𝑞^∗(0)𝑓₀𝑞(𝜃),𝜃∈[−1,0),𝐴𝑊−2Re𝑞^∗(0)𝑓₀𝑞(0)+𝑓₀(,𝜃=0,=𝐴𝑊+𝐻𝑧,𝑧,𝜃), (3.20) where 𝐻(𝑧,𝑧,𝜃)=𝐻₂₀(𝜃)𝑧²2+𝐻₁₁(𝜃)𝑧𝑧+𝐻₀₂(𝜃)𝑧²2+⋯. (3.21) Expanding the above series and comparing the coefficients, we obtain 𝐴−2𝑖𝜔₀𝜏₀𝐼𝑊₂₀(𝜃)=−𝐻₂₀(𝜃),𝐴𝑊₁₁(𝜃)=−𝐻₁₁(𝜃),…. (3.22) Noticing that 𝑞(𝜃)=1,𝛽,0^𝑇𝑒^{𝑖𝜔0𝜏0𝜃},𝑢_𝑡(𝜃)=𝑧𝑞(𝜃)+𝑧𝑞(𝜃)+𝑊(𝑧,𝑧,𝜃), (3.23) we have 𝑥(𝑡)=𝑧+𝑧+𝑊⁽¹⁾(𝑧,),𝑦(𝑡)𝑧,0=𝑧𝛽+𝑧𝛽+𝑊⁽²⁾(𝑧,),𝑥()𝑧,0𝑡−1=𝑧𝑒^{−𝑖𝜔0𝜏0}+𝑧𝑒^{𝑖𝜔0𝜏0}+𝑊⁽¹⁾(𝑧,),𝑦()𝑧,−1𝑡−1=𝑧𝛽𝑒^{−𝑖𝜔0𝜏0}+𝑧𝛽𝑒^{𝑖𝜔0𝜏0}+𝑊⁽²⁾(𝑧,),𝑛(𝑡)𝑧,−1=𝑊⁽³⁾(𝑧,𝑧,0), (3.24) where 𝑊⁽¹⁾(𝑧,𝑧,𝜃)=𝑊⁽¹⁾₂₀(𝜃)𝑧²2+𝑊⁽¹⁾₁₁(𝜃)𝑧𝑧+𝑊⁽¹⁾₀₂(𝜃)𝑧²2𝑊+⋯,₍₂₎(𝑧,𝑧,𝜃)=𝑊⁽²⁾₂₀(𝜃)𝑧²2+𝑊⁽²⁾₁₁(𝜃)𝑧𝑧+𝑊⁽²⁾₀₂(𝜃)𝑧²2+⋯, (3.25) and recalling that 𝑓₀=𝜏₀𝑥(𝑡)𝑛(𝑡)2−𝛼𝑦(𝑡)𝑛(𝑡)2𝛼𝑥(𝑡)𝑛(𝑡)2+𝑦(𝑡)𝑛(𝑡)2−1𝑇1+𝑝+𝑛(𝑡)𝑥²(𝑡)+𝑦²(𝑡), (3.26) we have 𝑔(𝑧,𝑧)=𝑞^∗(0)𝑓₀=𝜏₀𝑥(𝑡)𝑛(𝑡)(𝑡)𝑛(𝑡+2𝐷−𝛼𝑦)𝜇(𝑡)𝑛(𝑡)(𝑡)𝑛(𝑡=𝜏𝛼𝑥+𝑦)₀2𝐷(1+𝛼𝜈)𝑧+𝑧+𝑊⁽¹⁾₂₀(0)𝑧²2+𝑊⁽¹⁾₁₁(0)𝑧𝑧+𝑊⁽¹⁾₀₂(0)𝑧²2×𝑊+⋯₍₃₎₂₀(0)𝑧²2+𝑊⁽³⁾₁₁(0)𝑧𝑧+𝑊⁽³⁾₀₂(0)𝑧²2+(+⋯𝜈−𝛼)𝑧𝛽+𝑧𝛽+𝑊⁽²⁾₂₀(0)𝑧²2+𝑊⁽²⁾₁₁(0)𝑧𝑧+𝑊⁽²⁾₀₂(0)𝑧²2×𝑊+⋯₍₃₎₂₀(0)𝑧²2+𝑊⁽³⁾₁₁(0)𝑧𝑧+𝑊⁽³⁾₀₂(0)𝑧²2+⋯. (3.27) We can obtain the coefficients which will be used in determining the important quantities: 𝑔₂₀=𝑔₁₁=𝑔₀₂𝑔=0,₂₁=𝜏₀𝐷𝑊⁽³⁾₂₀(0)21−𝛼𝛽+𝛼𝜈+𝛽𝜈+𝑊⁽³⁾₁₁(0)1−𝛼𝛽+𝛼𝜈+𝛽𝜈. (3.28) We still need to compute 𝑊₂₀(𝜃) and 𝑊₁₁(𝜃) for 𝜃∈[−1,0). We have 𝐻(𝑧,𝑧,𝜃)=−𝑞^∗(0)𝑓₀𝑞(𝜃)−𝑞^∗(0)𝑓₀𝑞(𝜃)=−𝑔(𝑧,𝑧)𝑞(𝜃)−𝑔(𝑧,𝑧)𝑞(𝜃). (3.29) Comparing the coefficients about 𝐻(𝑧,𝑧,𝜃) gives that 𝐻₂₀(𝜃)=−𝑔₂₀𝑞(𝜃)−𝑔₀₂𝑞(𝜃)=0,𝐻₁₁=−𝑔₁₁𝑞(𝜃)−𝑔₁₁𝑞(𝜃)=0. (3.30) Then, from (3.22), we get 𝑊̇₂₀(𝜃)=2𝑖𝜔₀𝜏₀𝑊₂₀(𝜃),𝑊̇₁₁(𝜃)=0, (3.31) which implies that 𝑊₂₀(𝜃)=𝐸𝑒^{2𝑖𝜔0𝜏0𝜃},𝑊₁₁(𝜃)=𝐹, (3.32) where 𝐸,𝐹 are both three-dimensional vectors and can be determined by setting 𝜃=0 in 𝐻(𝑧,𝑧,𝜃). In fact, from 𝐻(𝑧,𝑧,0)=−2Re𝑞^∗(0)𝑓₀𝑞(0)+𝑓₀=−𝑔(𝑧,𝑧)𝑞(0)−𝑔(𝑧,𝑧)𝑞(0)+𝜏₀2𝑥(𝑡)𝑛(𝑡)(𝑡)𝑛(𝑡)(𝑡)𝑛(𝑡)(𝑡)𝑛(𝑡)−𝛼𝑦𝛼𝑥+𝑦−2𝑇1+𝑝+𝑛(𝑡)𝑥²(𝑡)+𝑦²(𝑡)=−𝑔(𝑧,𝑧)𝑞(0)−𝑔(𝑧,𝑧)𝑞(0)+𝜏₀2𝑊⁽¹⁾+𝑧+𝑧𝑊⁽³⁾−𝛼𝑊⁽²⁾+𝑧𝛽+𝑧𝛽𝑊⁽³⁾𝛼𝑊⁽¹⁾+𝑧+𝑧𝑊⁽³⁾+𝑊⁽²⁾+𝑧𝛽+𝑧𝛽𝑊⁽³⁾−2𝑇1+𝑝+𝑊⁽³⁾𝑊⁽¹⁾+𝑧+𝑧²+𝑊⁽²⁾+𝑧𝛽+𝑧𝛽², (3.33) we have 𝐻₂₀(0)=−𝑔₂₀𝑞(0)−𝑔₀₂𝑞(0)+𝜏₀20,0,−𝑇1+𝑝1+𝛽^2𝑇=𝜏₀20,0,−𝑇1+𝑝1+𝛽^2𝑇,𝐻₁₁(0)=−𝑔₁₁𝑞(0)−𝑔₁₁𝑞(0)+𝜏₀20,0,−𝑇1+𝑝1+𝛽^2𝑇=𝜏₀20,0,−𝑇1+𝑝1+𝛽^2𝑇. (3.34) It follows from (3.22) and the definition of 𝐴 that 𝜏₀𝐵𝑊₂₀(0)+𝜏₀𝐶𝑊₂₀(−1)=2𝑖𝜔₀𝜏₀𝑊₂₀(0)−𝐻₂₀(0),𝜏₀𝐵𝑊₁₁(0)+𝜏₀𝐶𝑊₁₁(−1)=−𝐻₁₁(0). (3.35) Combining the conditions above, we have 𝐵+𝑒^{−2𝑖𝜔0𝜏0}𝐶−2𝑖𝜔₀𝐼2𝐸=0,0,𝑇1+𝑝1+𝛽^2𝑇,()2𝐵+𝐶𝐹=0,0,𝑇1+𝑝1+𝛽^2𝑇, (3.36) which implies 𝐸=0,0,−21+2𝑖𝑇𝜔₀1+𝑝1+𝛽^2𝑇,𝐹=0,0,−21+𝑝1+𝛽². (3.37) Consequently, the above 𝑔₂₁ can be expressed by the parameters and delay in system (3.1). Thus, we can compute the following quantities: 𝑐₁(0)=𝑖2𝜔₀𝜏₀𝑔₂₀𝑔₁₁−2𝑔₁₁²−13𝑔₀₂²+𝑔₂₁2=−𝜏₀1+𝑝12𝐷1+2𝑖𝜔₀𝑇1+𝛽²1−𝛼𝛽+𝛼𝜈+𝛽𝜈+21+𝛽²1−𝛼𝛽+𝛼𝜈+𝛽𝜈,𝜇₂=−Re𝑐₁(0)Re𝜆^′𝜏₀,𝛽₂=2Re𝑐₁(0),𝑇₂=−Im𝑐₁(0)+𝜇₂Im𝜆^′𝜏₀𝜔₀𝜏₀, (3.38) which determine the properties of bifurcating periodic solutions at the critical value 𝜏₀. The direction and stability of Hopf bifurcation in the center manifold can be determined by 𝜇₂ and 𝛽₂, respectively. In fact, if 𝜇₂>0(𝜇₂<0), then the bifurcating periodic solutions are forward (backward); the bifurcating periodic solutions on the center manifold are stable (unstable) if 𝛽₂<0(𝛽₂>0); 𝑇₂ determines the period of the bifurcating periodic solutions: the period increases (decreases) if 𝑇₂>0(𝑇₂<0).

From the discussion in Section 2, we have known that (i)Re𝜆^′𝜏⁺_𝑗>0,(ii)Re𝜆^′𝜏⁻_𝑗𝑝<0,when2√<𝑘<1+𝛼²2√𝑝,>0,when𝑘>1+𝛼²2𝑝. (3.39) From Remark 2.5, we have the following results.

Theorem 3.1. For system (1.4), suppose that condition (H₂) is satisfied. Then, when 0≤𝑗≤𝑚, one has(i)the Hopf bifurcation at 𝐸(0,0,𝑝) when 𝜏=𝜏⁻_𝑗 is backward (forward) and the bifurcation periodic solutions are stable (unstable) if Re(𝑐₁(0))<0(>0);(ii)the Hopf bifurcation at 𝐸(0,0,𝑝) when 𝜏=𝜏⁺_𝑗 is forward (backward) and the bifurcation periodic solutions are stable (unstable) if Re(𝑐₁(0))<0(>0).Here 𝜏^±_𝑗 and 𝑚 are given in (ii) of Theorem 2.4.

4. Numerical Simulations

In this section, we will carry out numerical simulation on system (1.4). Set(A)  𝛼=9.42,𝑘=1,𝑝=1,𝑇=20.Clearly, (H₂) is satisfied. We have 𝜔₋≐3.845, 𝜔₊≐5.575 and 𝜏⁻₀≐0.5448,𝜏⁺₀≐0.7507,𝜏⁺₁≐1.8771,𝜏⁻₁≐2.1782⋯. (4.1) Thus, the equilibrium (0,0,1) is stable when 𝜏∈(𝜏⁻₀,𝜏⁺₀), and unstable when 𝜏∈[0,𝜏⁻₀)∪(𝜏⁺₀,∞). Furthermore, we get Re𝜆^′𝜏⁻₀≐−1.8411,Re𝜆^′𝜏⁺₀≐2.3133. (4.2) By the algorithm derived in Section 3, we can obtain Re𝑐₁(0)≐−0.7414,𝜇₂≐−0.1027,𝛽₂≐−1.4828 (4.3) at 𝜏=𝜏⁻₀, and Re𝑐₁(0)≐−4.7053,𝜇₂≐2.0340,𝛽₂≐−9.4106 (4.4) at 𝜏=𝜏⁺₀, respectively. These imply that the direction of Hopf bifurcations is backward when 𝜏=𝜏⁻₀, and forward when 𝜏=𝜏⁺₀, 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 𝜏−0≐0.5448, and the bifurcating periodic solutions are asymptotically stable, where 𝜏=0.54<0.5448 and the initial value is taken as (0.01,0.01,0.98).

(a)

(b)

(c)

(d)

(e)

Figure 2 

For system (1.4) with the data (A), the equilibrium 𝐸(0,0,1) is asymptotically stable when 𝜏∈(𝜏−0,𝜏+0), where 𝜏=0.6 and the initial value is taken as (0.01,0.01,0.98).

(a)

(b)

(c)

(d)

(e)

Figure 3 

For system (1.4) with the data (A), the Hopf bifurcation is forward at 𝜏+0≐0.7507, and the bifurcating periodic solutions are asymptotically stable, where 𝜏=0.77>𝜏+0 and the initial value is taken as (0.01,0.01,0.98).

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 𝑘 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 (H₂), the stability switch for 𝜏 occurs, which means that there exists a sequence values of 𝜏^±_𝑗 and an integer 𝑚 satisfying 0<𝜏⁻₀<𝜏⁺₀<𝜏⁻₁<⋯<𝜏⁻_𝑚<𝜏⁺_𝑚<𝜏⁺_𝑚+1, (5.1) such that the equilibrium 𝐸 is asymptotically stable when 𝜏∈∪^𝑚_𝑗=0(𝜏⁻_𝑗,𝜏⁺_𝑗), and unstable when 𝜏∈{0}∪(∪^𝑚_𝑗=0[𝜏⁺_𝑗−1,𝜏⁻_𝑗))∪(𝜏⁺_𝑚,∞), and the system undergoes a Hopf bifurcation at 𝜏=𝜏^±_𝑗, where 𝑗=1,2,….

As a result, the modulation of the coupling strength 𝑘 and the delay 𝜏 (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).