# Stability and Hopf Bifurcation Analysis of Coupled Optoelectronic Feedback Loops

Authors: Gang Zhu; Junjie Wei
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

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.


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## Body

## 1. Introduction

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]:
(1)dx1(t)dt=-(γ1+γ2)x1(t)-γ2y1(t)-βγ2cos2[x1(t-τ)+φ0],dy1(t)dt=γ1x1(t).
Here, x1(t) is the normalized voltage signal applied to the electrooptic modulator, τ is the feedback time delay, γ1 and γ2 are the filter low-pass and high-pass corner frequencies, β is the dimensionless feedback strength, they are all positive constants, and φ0 is the bias point of the modulator.Depending on the value of the feedback strengthβ and delay τ, 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
(2)dx2(t)dt=-(γ1+γ2)x2(t)-γ2y2(t)-βγ2cos2[kx1(t-τ)+(1-k)x2(t-τ)+φ0],dy2(t)dt=γ1x2(t),
for the receiver. In (2), k>0 denotes the coupling strength. We will find that with the variety of k, the dynamical behavior of the coupled system can be different, while the feedback strength β keeps the same value.The paper is organized as follows. In Section2, 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 β. 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(3)dx1(t)dt=-(γ1+γ2)x1(t)-γ2y1(t)-βγ2cos2[x1(t-τ)+φ0],dy1(t)dt=γ1x1(t),dx2(t)dt=-(γ1+γ2)x2(t)-γ2y2(t)-βγ2cos2[kx1(t-τ)+(1-k)x2(t-τ)+φ0],dy2(t)dt=γ1x2(t).It is easy to see thatE(0,-βcos2φ0,0,-βcos2φ0) is the only equilibrium of system (3). Linearizing system (3) around E and denote δ=sin2φ0, we get the linearization system
(4)dx1(t)dt=-(γ1+γ2)x1(t)-γ2y1(t)+βδγ2x1(t-τ),dy1(t)dt=γ1x1(t),dx2(t)dt=-(γ1+γ2)x2(t)-γ2y2(t)+kβδγ2x1(t-τ)+(1-k)βδγ2x2(t-τ),dy2(t)dt=γ1x2(t),
and the characteristic equation of system (4)
(5)[λ2+(γ1+γ2)λ+γ1γ2-(1-k)βδγ2λe-λτ]×[λ2+(γ1+γ2)λ+γ1γ2-βδγ2λe-λτ]=0,
which is equivalent to
(6)λ2+(γ1+γ2)λ+γ1γ2-βδγ2λe-λτ=0,(7)λ2+(γ1+γ2)λ+γ1γ2-(1-k)βδγ2λe-λτ=0.Notice that whenβ=0, (5) becomes
(8)[λ2+(γ1+γ2)λ+γ1γ2]2=0,
whose roots are
(9)λ1,2=-γ1,λ3,4=-γ2.
So, we have the following lemma.Lemma 1.
The equilibriumE0(0,-βcos2φ0,0,-βcos2φ0) is asymptotically stable when β=0.Next, we regardβ as the bifurcation parameter to investigate the distribution of roots of (6) and (7).Letλ=iω  (ω>0) be a root of (6) and substituting λ=iω into (6), separating the real and imaginary parts yields
(10)-ω2+γ1γ2=βδγ2ωsinωτ,ω(γ1+γ2)=βδγ2ωcosωτ.
Then, we can get
(11)tanωτ=-ω2+γ1γ2ω(γ1+γ2).
Hence, (11) has a sequence of roots {ωj}j≥0 (see Figure 1), and
(12)ωj∈{(2jπτ,2jπ+π/2τ),ωj2<γ1γ2,(2jπ+3π/2τ,2(j+1)πτ),ωj2>γ1γ2,j=0,1,2,….Figure 1
The points of intersection off1=tanωτ and f2=(-ω2+γ1γ2)/ω(γ1+γ2), when γ1=0.1, γ2=2.5,  τ=1.5.Define(13)βj=γ1+γ2δγ2cosωjτ.
Then, (ωj,βj) is the solution of (10).From (10), we know that
(14)β2=1δ2γ22[(γ1γ2ω-ω)2+(γ1+γ2)2],
which gives that
(15)dβdω=1βδ2γ22ω3(ω2+γ1γ2)(ω2-γ1γ2).
From Figure 1, we know that ωj→∞ when j→∞, which means that ωj2>γ1γ2; furthermore, β is increasing with respect to ω, when j is sufficiently big.Reorder the set{βj} such that β0=min{βj} and ωj is correspondent of βj(j=0,1,2,…). Then, we have the following lemma.Lemma 2.
There exists a sequence values ofβ denoted by
(16)0<β0<β1<⋯,
such that (6) has a pair of imaginary roots ±iωj when β=βj(j=0,1,2,…), where βj is defined by (13), and ωj is the root of (11).Let(17)λ(τ)=α(β)+iω(β)
be the root of (6) satisfying α(βj)=0 and ω(βj)=ωj. We have the following conclusion.Lemma 3.
α
′
(
β
j
)
>
0.Proof.
Substitutingλ(β) into (6) and taking the derivative with respect to β, it follows that
(18)2λdλdβ+(γ1+γ2)dλdβ-δγ2λe-λτ-βδγ2e-λτdλdβ+τβδγ2λe-λτdλdβ=0.
Therefore, noting that βδγ2λe-λτ=λ2+(γ1+γ2)λ+γ1γ2, we have
(19)dλdβ=1βλ3+(γ1+γ2)λ2+γ1γ2λλ2-γ1γ2+τ[λ3+(γ1+γ2)λ2+γ1γ2λ],
and by a straight computation, we get
(20)α′(βj)=ωj2βjΔ[τωj2(γ12+γ22)+(ωj2+γ1γ2)(γ1+γ2)+τωj4+τγ12γ22]>0,
where
(21)Δ=[(ωj2+γ1γ2)+τωj2(γ1+γ2)]2+[τωj(ωj2-γ1γ2)]2.As to (7), it can be easily found that -γ1,-γ2 are two negative roots when k=1, so, next, we only focus on (7) with k≠1.Letλ=iϖ(ϖ)>0 be a root of (7). Using the same method above, we get
(22)-ϖ2+γ1γ2=(1-k)βδγ2ϖsinϖτ,(γ1+γ2)ϖ=(1-k)βδγ2ϖcosϖτ,(23)tanϖτ=-ϖ2+γ1γ2ϖ(γ1+γ2).Then, when0<k<1, (23) has a sequence of roots {ϖj}j≥0, which are the same as those of (11).Whenk>1, (23) has a sequence of roots {ϖj}j≥0, and
(24)ϖj∈{((2j+1)πτ,(2j+1)π+π/2τ),ϖj2<γ1γ2,(2jπ+π/2τ,(2j+1)πτ),ϖj2>γ1γ2,j=0,1,2,….Define(25)β-j=γ1+γ2(1-k)δγ2cosϖτ.
Then, (ϖj,β-j) is the solution of (22).Repeat the previous process, we have(26)dβ-dϖ=1(1-k)2βδ2γ22ϖ3(ϖ2+γ1γ2)(ϖ2-γ1γ2).Reorder the set{β-j} such that β-0=min{β-j} and ϖj is correspondent of β-j(j=0,1,2,…).Lemma 4.
There exists a sequence values ofβ- denoted by
(27)0<β-0<β-1<⋯,
such that (7) has a pair of imaginary roots ±iϖj when β-=β-j(j=0,1,2,…), where β-j is defined by (25), and ϖj is the root of (23).Let(28)λ(τ)=α(β-)+iϖ(β-)
be the root of (7) satisfying α(β-j)=0, ϖ(β-j)=ϖj. Then, similar to the proof of Lemma 3, we have the following conclusion.Lemma 5.
α
′
(
β
-
j
)
>
0.Compareβj,  β-j and reorder the set {βj} and {β-j} and remove the “−” of β-j, such that
(29)0<β0<β1<⋯,
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).Theorem 6.
For system(3), the equilibrium E is asymptotically stable when β∈[0,β0) and unstable when β∈(β0,+∞); system (3) undergoes a Hopf bifurcation at E when β=βj, j=0,1,2,…, where βj are defined by (13) or (25).


## 3. The Direction and Stability of the Hopf Bifurcation

In Section2 we obtained some conditions under which system (3) undergoes the Hopf bifurcation at some critical values of β. 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].MoveE(0,-βcos2φ0,0,-βcos2φ0) to the origin O(0,0,0,0) and denote δ=sin2φ0,  ρ=cos2φ0, then system (3) can be written as the form
(30)dx1(t)dt=-(γ1+γ2)x1(t)-γ2y1(t)+βδγ2x1(t-τ)+βγ2ρφ0x12(t-τ)-23βδγ2x13(t-τ)+O(4),dy1(t)dt=γ1x1(t),dx2(t)dt=-(γ1+γ2)x2(t)-γ2y2(t)+βγ2[∫kδx1(t-τ)+(1-k)δx2(t-τ)+k2ρx12(t-τ)+2k(1-k)ρx1(t-τ)x2(t-τ)+(1-k)2ρx22(t-τ)-23k3δx13(t-τ)-2k2(1-k)δx12(t-τ)x2(t-τ)-2k(1-k)2δx1(t-τ)x22(t-τ)-23(1-k)3δx23(t-τ)∫]+O(4),dy2(t)dt=γ1x2(t).Clearly, the phase space is𝒞=𝒞([-τ,0],ℝ4). For convenience, let
(31)β*∈{βj}∪{β-j},
and β=β*+μ, μ∈ℝ. From the analysis above we know that μ=0 is the Hopf bifurcation value for system(30). Let iω* be the root of the characteristic equation associate with the linearization of system (30) when β=β*. For ϕ=(ϕ1,ϕ2,ϕ3,ϕ4)∈𝒞, let
(32)Lμ(ϕ)=Bϕ(0)+Cϕ(-τ),
where
(33)B=(-(γ1+γ2)-γ200γ100000-(γ1+γ2)-γ200γ10),C=(βδγ20000000kβδγ20(1-k)βδγ200000).
By the Rieze representation theorem, there exists a 4×4 matrix, η(θ,μ)(-τ≤θ≤0), whose elements are of bounded variation functions such that
(34)Lμ(ϕ)=∫-τ0‍dη(θ,μ)ϕ(θ),ϕ∈𝒞.In fact, we can choose(35)η(θ,μ)={B,θ=0,0,θ∈(-τ,0)-C,θ=-τ.
Then, (30) is satisfied.Forϕ∈𝒞, define the operator A(μ) as
(36)A(μ)ϕ(θ)={dϕ(θ)dθ,θ∈[-τ,0),∫-τ0dη(t,μ)ϕ(t),θ=0,
and R(μ)ϕ as
(37)R(μ)ϕ(θ)={0,θ∈[-τ,0),f(μ,ϕ),θ=0,
where (38)f(μ,ϕ)=β*γ2(ρϕ12(-τ)-23δϕ13(-τ)+O(4)0k2ρϕ12(-τ)+k(1-k)ρϕ1(-τ)ϕ3(-τ)+(1-k)2ρϕ32(-τ)-23k3δϕ13(-τ)-2k2(1-k)δϕ12(-τ)ϕ3(t-τ)-2k(1-k)2δϕ1(-τ)ϕ32(-τ)-23(1-k)3δϕ33(-τ)+O(4)0).Then, system (30) is equivalent to the following operator equation:
(39)u˙t=A(μ)ut+R(μ)ut,
where u(t)=(x1(t),y1(t),x2(t),y2(t))T,  ut=u(t+θ), for θ∈[-τ,0].Forψ∈𝒞1([0,τ],ℝ4), define
(40)A*ψ(s)={-dψ(s)ds,s∈(0,τ],∫-τ0ψ(-ξ)dη(ξ,0),s=0.Forϕ∈𝒞[-τ,0] and ψ∈𝒞[0,τ], define the bilinear form
(41)〈ψ(s),ϕ(θ)〉=ψ-(0)ϕ(0)-∫-τ0‍∫0θ‍ψ-(ξ-θ)dη(θ)ϕ(ξ)dξ,
where η(θ)=η(θ,0). Then, A(0) and A* are adjoint operators.Letq(θ) and  q*(s) be eigenvectors of A(0) and A* associated to iω* and -iω*, respectively. It is not difficult with verify that
(42)q(θ)=(1,γ1iω*,1,γ1iω*)Teiω*θ,q*(s)=1D-(1,γ2iω*,1,γ2iω*)eiω*s,
where
(43)D=2+2γ1γ2ω*2+2β*δγ2τe-iω*τ.
Then, 〈q*(s),q(θ)〉=1,  〈q*(s),q-(θ)〉=0.Letut be the solution of (39) and define
(44)z(t)=〈q*,ut〉,W(t,θ)=ut(θ)-2Re{z(t)q(θ)}.
On the center manifold 𝒞0, we have
(45)W(t,θ)=W(z(t),z-(t),θ),
where
(46)W(z,z-,θ)=W20z22+W11zz-+W02z-22+⋯,z and z- are local coordinates for center manifold 𝒞0 in the direction of q* and q-*. Note that W is real if ut is real. We only consider real solutions.For solutionut in 𝒞0, since μ=0, we have
(47)z˙(t)=iω*z+〈q*(θ),f(0,W+2Re{z(t)q(θ)})〉=iω*z+q-*(0),f(0,W(z,z-,0)+2Re{z(t)q(0)})=iω*z+q-*(0)f0(z,z-).
We rewrite this equation as
(48)z˙(t)=iω*z+g(z,z-),
where
(49)g(z,z-)=g20z22+g11zz-+g02z-22+g21z2z-2⋯.
By (39) and (48), we have
(50)W˙=u˙t-z˙q-z-˙q-={AW-2Re{q-*(0)f0q(θ)},θ∈[-τ,0),AW-2Re{q-*(0)f0q(0)}+f0,θ=0,=AW+H(z,z-,θ),
where
(51)H(z,z-,θ)=H20(θ)z22+H11(θ)zz-+H02(θ)z-22+⋯.
Expanding the above series and comparing the coefficients, we obtain
(52)(A-2iω*I)W20(θ)=-H20(θ),AW11(θ)=-H11(θ),….
Notice that(53)
q
(
θ
)
=
(
1
,
γ
1
i
ω
*
,
1
,
γ
1
i
ω
*
)
T
e
i
ω
*
θ
,
u
t
(
θ
)
=
z
q
(
θ
)
+
z
-
q
-
(
θ
)
+
W
(
z
,
z
-
,
θ
)
,
where
(54)W(i)(z,z-,θ)=W20(i)(θ)z22+W11(i)(θ)zz-+W02(i)(θ)z-22+⋯,i=1,2,3,4.
Combing (38) and by straightforward computation, we can obtain the coefficients which will be used in determining the important quantities:
(55)g20=2β*γ2ρDe-2iω*τ(k2-k+2),g11=2β*γ2ρD(k2-k+2),g02=2β*γ2ρDe2iω*τ(k2-k+2),g21=2β*γ2D{∑iiρ(eiω*τW20(1)(-τ)+2e-iω*τW11(1)(-τ))+k2ρ(eiω*τW20(1)(-τ)+2e-iω*τW11(1)(-τ))+k(1-k)ρ×(e-iω*τW11(3)(-τ)+eiω*τW20(3)(-τ)2+eiω*τW20(1)(-τ)2+e-iω*τW11(1)(-τ))+(1-k)2ρ(ss2e-iω*τW11(3)(-τ)+eiω*τW20(3)(-τ)ss)-4δe-iω*τ∑ii}.
We still need to compute W20(θ) and W11(θ), for θ∈[-τ,0). We have
(56)H(z,z-,θ)=-q-*(0)f0q(θ)-q*(0)f-0q-(θ)=-g(z,z-)q(θ)-g-(z,z-)q-(θ).
Comparing the coefficients about H(z,z-,θ) gives that
(57)H20(θ)=-g20q(θ)-g-02q-(θ),H11=-g11q(θ)-g-11q-(θ).
Then, from (52), we get
(58)W˙20(θ)=2iω*W20(θ)+g20q(θ)+g-02q-(θ),W˙11(θ)=g11q(θ)+g-11q-(θ),
which implies that
(59)W20(θ)=g20q(0)-iω*eiω*θ+g-02q-(0)-3iω*e-iωj*θ+Ee2iωj*θ,W11(θ)=g11q(0)iω*eiω*θ+g-11q-(0)-iω*e-iω*θ+F.
Here, E and F are both four-dimensional vectors and can be determined by setting θ=0 in H(z,z-,θ). In fact, from (38) and
(60)H(z,z-,0)=-2Re{q-*(0)f0q(0)}+f0,
we have
(61)H20(0)=-g20q(0)-g-02q-(0)+2β*γ2ρe-2iω*τ(1,0,k2-k+1,0)T,H11(0)=-g11q(0)-g-11q-(0)+2β*γ2ρ(1,0,k2-k+1,0)T.
It follows from (52) and the definition of A that
(62)β*BW20(0)+β*CW20(-τ)=2iω*W20(0)-H20(0),β*BW11(0)+β*CW11(-τ)=-H11(0),
which implies that
(63)E=(B+e-2iω*τC-2iω*I)-1×[∑iiB(g20q(0)iω*+g-02q-(0)3iω*)+C(g20q(0)iω*e-iω*τ+g-02q-(0)3iω*eiω*τ)+1β*(g20q(0)+g-02q-(0))×2β*γ2ρe-2iω*τ(1,0,k2-k+1,0)T∑ii],F=(B+C)-1[∑B(g11q(0)-iω*+g-11q-(0)iω*)+C(g11q(0)-iω*e-iω*τ+g-11q-(0)iω*eiω*τ)+1β*(g11q(0)+g-11q-(0))-2β*γ2ρ(1,0,k2-k+1,0)T∑].
Consequently, the above g21 can be expressed by the parameters and delay in system (30). Thus, we can compute the following quantities:
(64)c1(0)=i2ω*(g20g11-2|g11|2-13|g20|2)+g212,μ2=-Rec1(0)Reλ′(β*),β2=2Rec1(0),T2=-Imc1(0)+μ2Imλ′(β*)ω*,
which determine the properties of bifurcating periodic solutions at the critical value τ0. The direction and stability of the 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); and T2 determines the period of the bifurcating periodic solutions: the period increases (decreases) if T2>0(T2<0).From the discussion in Section2, we have known that Reλ′(βj)>0; therefore; we have the following result.Theorem 7.
The direction of the Hopf bifurcation for system (3) at the equilibrium E(0,-βcos2φ0,0,-βcos2φ0) when β=β* is forward (backward), and the bifurcating periodic solutions on the center manifold are stable (unstable) if Re(c1(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 β=β0.


## 4. Numerical Simulations

In this section, we will carry out numerical simulations on system (3) at special values of β. We choose a set of data as follows:
(65)γ1=0.1,γ2=2.5,φ0=π4,τ=1.5,
which are the same as those in [1]. Then, δ=1,ρ=0.Then, we can obtain(66)ω0=·0.2225,ω1=·3.5677,…,ϖ0=·1.7294,ϖ1=·5.5531,…,β0=·1.1008,β1=·1.7434,…,β-0=·1.3534,β-1=·2.5235,…,k=1.9,β-0=·0.6093,β-1=·1.1356,…,k=3.
From the analysis in Section 2, we know that β(β-) is increasing with respect to ω(ϖ) when ω(ϖ)>γ1γ2, which means that
(67)β0=min{βj},β-0=min{β-j},j=0,1,2,…,
that is, β0(β-0) is the first critical value at which system (3) undergoes a Hopf bifurcation.Whenk=1.9, by the previous results, it follows that
(68)λ′(β0)=·0.2440-0.0491i,c1(0)=·-0.5373+0.1082i,μ2=·2.2020,β2=·-1.0746,T2=-0.0018.Hence, we arrive at the following conclusion: the equilibriumE is asymptotically stable when β∈[0,1.1008) and unstable when β∈(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).γ
1
=
0.1, γ2=2.5, τ=1.5, k=1.9, which means that condition (H1) holds, and β=0.7<β0. The initial value is (0.1,-0.5,0.1,-0.5).
(a)
(b)
(c)
(d)γ
1
=
0.1, γ2=2.5, τ=1.5, k=1.9, which means that condition (H1) holds, and β=1.2>β0. The initial value is (0.1,-0.5,0.1,-0.5).
(a)
(b)
(c)
(d)Whenk=3, we can get
(69)λ′(β-0)=·0.9004+0.0924i,c1(0)=·-1.9116+0.7930i,μ2=·2.1231,β2=·-3.8232,T2=-0.5720.Then, we have the following: the equilibriumE is asymptotically stable when β∈[0,0.6093), and unstable when β∈(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).γ
1
=
0.1, γ2=2.5, τ=1.5, k=3, which means that condition (H2) holds, and β=0.6<β-0. The initial value is (0.1,-0.5,0.1,-0.5).
(a)
(b)
(c)
(d)γ
1
=
0.1, γ2=2.5, τ=1.5, k=3, which means that condition (H2) holds, and β-0<β=0.7<β0. The initial value is (0.1,-0.5,0.1,-0.5).
(a)
(b)
(c)
(d)γ
1
=
0.1, γ2=2.5, τ=1.5, k=3, which means that condition (H2) holds, and β=1.2>β0>β-0. The initial value is (0.1,-0.5,0.1,-0.5).
(a)
(b)
(c)
(d)


## 5. 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 β and delay τ, 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 strengthk, even if all other parameters keep the same, the dynamical behavior can change greatly. In fact, it is clear that the first two equations, x1(t) and y1(t) are uncoupled with equations x2(t) and y2(t), so system (1) are independent of (2), which means that coupling strength k does not appear in (1). The characteristic equation of (1) has the same form as (6), so the first critical value β0 is independent of k. The analysis of characteristic equation (7) shows that the value of k can affect the first critical value β-0 definitely. And we draw a conclusion that when k is in an interval, in which β0<β-0 holds, solutions of system (1) and (2) keep synchronous; when k belongs to the interval, in which β-0<β0 holds, solutions of system (1) and (2) can also keep synchronous with β<β-0, while they lose their synchronization when β>β-0, no matter whether β<β0 or not.As a result, the modulation of the coupling strengthsk together with the feedback strength β would be an efficient and an easily implementable method to control the behavior of the coupled chaotic oscillators.



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*Source: 918943-2013-03-13.xml*