Bifurcations for A Single Mode Laser Model with Time Delay in Frequency Domain Methods

Bifurcations for A Single Mode Laser Model with

Changjin Xua,∗, Peiluan Lib

a _{Guizhou Key Laboratory of Economics System Simulation, Guizhou College of Finance and Economics,}

Guiyang 550004, PR China Email: [email protected]

b _{Department of Mathematics and Statistics, Henan University of Science and Technology,}

Luoyang 471003, PR China Email: [email protected]

Abstract— In this paper, the dynamic behavior of a single mode laser model with delay is investigated via frequency domain approach. By choosing the delayτ as a bifurcation parameter, we show that Hopf bifurcation can occur when τ passes a sequence of critical values. This means that a family of periodic solutions bifurcate from the equilibrium when the bifurcation parameter exceeds a critical value. Some numerical simulation results are consistent with the theoretical ones. The approach used in this paper is an excel-lent supplement of previous known ones of Hopf bifurcation analysis.

Index Terms— single mode laser model, Hopf bifurcation, stability, frequency domain approach, Nyquist criterion

I. INTRODUCTION

In optics, laser is a typical example which produces the phenomenon of self-organization of bistability, oscillation and chaos[1]. It is currently one of the focus of research in laser field. In recent year, a great many laser models that obtain self-organization phenomena have been proposed and studied. In 1998, Lu et al.[2] investigated the chaotic behavior of the following single mode laser system:

⎧ ⎪ ⎨ ⎪ ⎩

dx1(t)

dt =−ax1(t) +x2(t), dx2(t)

dt =−bx2(t) +x1(t)x3(t), dx3(t)

dt =c−x3(t)−x1(t)x2(t),

(1)

where x₁ denotes driving of atoms by electromagnetic field,x2is dipole moment of atom,x3denotes population inversion. The specific meaning of parameters for model (1), one can see [2]. Considering that there is a certain time delay during the input signal and the transmission signals, the dynamic behavior of the system not only is affected by the current state of the system, but also the past state of the system, i.e., there exists inherent lag in the system. Based on this view of point, in this paper, we will investigate the following single mode laser system

TThis work is supported by National Natural Science Foundation of China (No.60902044), the Doctoral Foundation of Guizhou College of Finance and Economics(2010) and the Science and technology Program of Hunan Province(No.2010FJ6021).

with delay: ⎧ ⎪ ⎨ ⎪ ⎩

dx1(t)

dt =−ax1(t) +x2(t), dx2(t)

dt =−bx2(t) +x1(t)x3(t−τ), dx3(t)

dt =c−x3(t−τ)−x1(t)x2(t).

(2)

In this paper, we investigate Hopf bifurcation of the sys-tem (2). It must be pointed out that many of the early work on Hopf bifurcation of the delayed differential equations is used the state-space formulation for delayed differen-tial equations, known as the “time domain” approach. Recently, there is another interesting approach, that is the frequency domain approach, which was initiated and developed by Allwright[3], Mees and Chua[4], Moiola and Chen[5,6] for studying delayed differential equations. One can see [7-9]. This alternative representation applies the familiar engineering feedback systems theory and methodology: an approach described in the “frequency domain”—the complex domain after the standard Laplace transforms have been taken on the state-space system in the time domain. This new methodology has some advan-tages over the classical time-domain methods. A typical one is its pictorial characteristic that utilizes advanced computer graphical capabilities thereby bypassing quite a lot of profound and difficult mathematical analysis.

In this paper, we will devote our attention to finding the Hopf bifurcation point for models (2) by means of the frequency-domain approach. It is found that if the coefficient τ is used as a bifurcation parameter, then Hopf bifurcation occurs for the model (2). This means that a family of periodic solutions bifurcates from the equilibrium when the bifurcation parameter exceeds a critical value. Some numerical simulations are given to justify the theoretical analysis results. To the best of our knowledge, it is the first time to deal with the research of Hopf bifurcation of the model (2) by the frequency-domain approach.

a critical value. In section 3, some numerical simulation are carried out to verify the correctness of theoretical analysis result. Finally, some conclusions and discussions are given in section 4.

II. EXISTENCE OFHOPFBIFURCATION

In model (2), we assume that the following condition

(H₎ c > ab.

holds.

It is easy to see that system (2) has a unique positive equilibriumE_∗(x∗₁, x∗₂, x∗₃₎,where,

x∗₁₌

c−ab a , x

∗

2=ax∗1, x∗3=ab.

We can rewrite the nonlinear system (2) in a matrix form as

dx₍t₎

dt =Ax(t) +H(x), (3)

wherex_{= (}x₁₍t₎, x₂₍t₎, x₃₍t₎₎T_,

A₌ ⎛

⎝ −₀a ₋1b 0₀

0 0 0

⎞ ⎠,

H₍x_{) =} ⎛

⎝ x₁₍t₎x₃0₍t−τ₎ c−x₃₍t−τ₎−x₁₍t₎x₂₍t₎

⎞ ⎠.

Choosing the coefficientτ as a bifurcation and introduc-ing a “state-feedback control”u₌g₍y₍t−τ_);τ₎,where

y₍t_{) = (}y₁₍t₎, y₂₍t₎, y₃₍t₎₎T_{, we obtain a linear system}

with a non-linear feedback as follows ⎧

⎨ ⎩

dx

dt =Ax+Bu,

y₌−Cx,

u₌g₍y₍t−τ_);τ₎,

(4)

where

B₌C₌ ⎛

⎝ 1 0 0_{0 1 0}

0 0 1

⎞ ⎠,

u_[g₍y−τ₎, τ_{] =} ⎛

⎝ y₁₍t₎y₃0₍t−τ₎ c−y₃₍t−τ₎−y₁₍t₎y₂₍t₎

⎞ ⎠.

Next, taking Laplace transform on (4), we obtain the standard transfer matrix of the linear part of the system:

G₍s_;τ_{) =}C_[sI−A_]−1B.

Then

G₍s_;τ_{) =} ⎛ ⎝

1

s+a (s+a)(s+b)1 0

0 _s+b1 0

0 0 1_s

⎞

⎠. ₍₅₎

If this feedback system is linearized about the equilibrium y ₌−C₍x∗₁, x∗₂, x∗₃₎T_{, then the Jacobian of (5) is given} by

J₍τ_{) =} ∂g ∂y

y=˜y=−C(x∗

1,x∗2,x∗3)T

=

⎛

⎝ x0∗₃ 0₀ x∗₁e0−sτ −x∗₂ −x∗₁ −e−sτ

⎞

⎠. ₍₆₎

Set

h₍λ, s_;τ_{) = det}|λI−G₍s_;τ₎J₍τ₎|

= λ

λ2−

e−sτ

s −

x∗₃

(s₊a₎₍s₊b₎

λ

+(x

∗

1x∗2−x∗3)e−sτ

s₍s₊a₎₍s₊b₎ +

(x∗₁₎2e−sτ

s₍s₊b₎

= 0.

Then, we obtain the following results by applying the generalized Nyquist stability criterion withs₌iω.

Lemma 2.1. [5] If an eigenvalue of the corresponding

Jacobian of the nonlinear system, in the time domain,

assumes a purely imaginary value iω₀ at a particular

τ ₌τ₀, then the corresponding eigenvalue of the constant

matrix G₍iω_0;τ₀₎J₍τ₀₎ in the frequency domain must

assume the value−_{1 +}i₀at τ ₌τ₀.

To apply Lemma 2.1, let ˆλ ₌ λˆ₍iω_;τ₎ be the eigenvalue of G₍iω_;τ₎J₍τ₎ that satisfies λˆ₍iω0;τ0) =

−1 + 0i. Thenh₍−₁, iω_0;τ_{0) =}

−

1 +

e−iω0τ0

iω₀ −

x∗₃

(iω₀₊a₎₍iω₀₊b₎

+ (x

∗

1x∗2−x∗3)e−iω0τ0

iω₀₍iω₀₊a₎₍iω₀₊b₎+

(x∗₁₎2e−iω0τ0

iω₀₍iω₀₊b₎

= 0.

Separating the real and imaginary parts, we obtain

A_1cosω₀τ₀₊A_2sinω₀τ_{0= (}a₊b₎ω₀2, ₍₇₎

A_2cosω₀τ₀−A_1sinω₀τ_{0= (}ω2₀−x∗₃−ab₎ω₀, ₍₈₎

where

A₁₌ab−ω₀2₊x∗₁x₂∗−x∗₃₊a₍x∗₁₎2,

A_{2= (}a₊b₊x∗₁₎ω₀. (9)

Thus, we have

A2₁₊A2_{2= [(}a₊b₎ω₀2_]2_{+ [(}ω2₀−x∗₃−ab₎ω_0]2

which leads to

ω6₀₊B1ω₀4₊B2ω₀2₊B3= 0, ₍₁₀₎

where

B_{1 =} a2₊b2−₂x∗₃−₁,

B_{2 = 2(}ab₊x∗₁x∗₂−x∗₃₊a₍x∗₁₎2₎−₍a₊b₊x∗₁₎2,

B_{3 =} −_[ab₊x∗₁x∗₂−x∗₃₊a₍x∗₁₎2_]2.

By (10), we can compute the value of ω₀ by means of Matlab software. Then from (10), we obtain

τ₀₌ 1 ω0

arccosA1(a+b)ω

2

0+A2(ω02−x∗3−ab)ω0

A2₁₊A2₂

+2kπ

Theorem 2.1. ( Existence of Hopf bifurcation

param-eter ) For system (2), ifω₀is positive real roots of (10),

then then Hopf bifurcation point of system (2) is

τ₀₌ 1 ω₀

arccosA1(a+b)ω

2

0+A2(ω20−x∗3−ab)ω0

A2₁₊A2₂

+2kπ

(k_{= 0},₁,₂,· · ·₎,

where,A₁, A₂ are defined by (9), respectively.

III. NUMERICALEXAMPLES

In this section, we present some numerical investiga-tions to verify the correctness of our analysis. As an example, we consider the following system:

⎧ ⎪ ⎨ ⎪ ⎩

dx1(t)

dt =−0.2x1(t) +x2(t), dx2(t)

dt =−0.3x2(t) +x1(t)x3(t−τ), dx3(t)

dt = 0.6−x3(t−τ)−x1(t)x2(t).

(12)

By Theorem 2.1, we obtainτ₀≈₀.₃₄. Let τ _{= 0}.₃₂, as shown in Figs.1-6, whenτ < τ₀ ≈₀.₃₄, system (12) is asymptotically stable. Letτ _{= 0}.₃₅, then the figures of numerical simulations are Figs.7-12. Thus we conclude that whenτ > τ₀≈₀.₃₄, system (12) undergoes a Hopf bifurcation occurs near the positive equilibrium. Therefore τ₀≈₀.₃₄is a supercritical Hopf bifurcation point.

0 50 100 150 200

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

t

x1

(t)

Fig.1

0 50 100 150 200

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

t

x2

(t)

Fig.2

0 50 100 150 200

−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

t

x3

(t)

Fig.3

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

x₁(t)

x2

(t)

Fig.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 −0.4

−0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

x₂(t)

x3

(t)

Fig.5

0.5 1

1.5 2

0 0.5 1 −0.4 −0.2 0 0.2 0.4 0.6

x1(t) Fig.6

x₂(t)

x3

(t)

0 50 100 150 200 1.1

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

t

x1

(t)

Fig.7

0 50 100 150 200

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

t

x2

(t)

Fig.8

0 50 100 150 200

−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

t

x3

(t)

Fig.9

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

x₁(t)

x2

(t)

Fig.10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 −0.4

−0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

x₂(t)

x3

(t)

Fig.11

0.5 1

1.5 2

0 0.5 1 −0.4 −0.2 0 0.2 0.4 0.6

x₁(t) Fig.12

x2(t)

x3

(t)

Fig.7-Fig.12. Numerical simulations of system (12) with τ _{= 0}.₃₅> τ₀≈₀.₃₄. Hopf bifurcation occurs from the positive equilibrium. The initial value is (0.5, 0.5, 0.5).

0 50 100 150 200 250 300

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

t

x1

(t)

Fig.13

0 50 100 150 200 250 300

−1.5 −1 −0.5 0 0.5 1 1.5

t

x2

(t)

0 50 100 150 200 250 300 −1.5

−1 −0.5 0 0.5 1 1.5 2

t

x3

(t)

Fig.15

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 −1.5

−1 −0.5 0 0.5 1 1.5 2

x1(t)

x2

(t)

Fig.16

−1.5 −1 −0.5 0 0.5 1 1.5

−1.5 −1 −0.5 0 0.5 1 1.5 2

x₂(t)

x3

(t)

Fig.17

−4 −2

0 2

4

−2 −1 0 1 2 −2 −1 0 1 2

x₁(t) Fig.18

x2(t)

x3

(t)

Fig.13-Fig.18. Numerical simulations of system (12) with τ _{= 0}.₅₅. Complicated dynamical phenomenon occurs The initial value is (0.5, 0.5, 0.5).

IV. CONCLUSION AND DISCUSSIONS

In this paper, we investigated a single mode laser model with time delay. By choosing the coefficient τ as a bifurcating parameter and analyzing the associating

characteristic equation. It is found that a Hopf bifur-cation occurs when the bifurcating parameter τ passes through a critical value. With the increase of the delay τ, system (12) will display the complicated dynamical behaviors(see Fig.13-Fig.18). Considering computational complexity, the direction and the stability of the bifurcat-ing periodic orbits for system (2) have not been studied. It is beyond the scope of the present paper and will be further investigated elsewhere in the future. In fact, the direction and the stability of the bifurcating periodic orbits can be studied by drawing the amplitude locus L₁ and the locus ˆλ₍iω₎ in a neighborhood of the Hopf bifurcation point[6]. Parameterσ was used to determine the direction of Hopf bifurcation: if σ > ₀, the Hopf bifurcation is supercritical; ifσ <₀, the Hopf bifurcation is subcritical, but ifσ_{= 0}, we cannot decide the direction of the bifurcating periodic orbits by using the L₁ and the locus λˆ₍iω₎[5]. In this case, we must resort to an more advanced algorithm including the forth-, sixth-, and even eighth-order harmonic balance approximations, i. e., the amplitude locus ₂, L₃, L₄, respectively, to find the corresponding solutions as carried out by Moiola and Chen[5]. The fundamental equations about the amplitude loci2, L3, L4 are

ˆ

λ₍iω_{) =} −_{1 +}ξ_1(˜ω₎θ2₊ξ_2(˜ω₎θ4,

ˆ

λ₍iω_{) =} −_{1 +}ξ_1(˜ω₎θ2₊ξ_2(˜ω₎θ4₊ξ_3(˜ω₎θ6,

ˆ

λ₍iω_{) = 1 +}ξ1(˜ω₎θ2₊ξ2(˜ω₎θ4₊ξ3(˜ω₎θ6₊ξ4(˜ω₎θ8.

By applying these high-order Hopf bifurcating formulas, we can expect to obtain more accurate results and the global bifurcating behaviors. Since this task is computa-tionally intensive, it will be further investigated elsewhere in the future.

ACKNOWLEDGMENT

We thank the reviewers for their value comments that lead to truly significant improvement of the manuscript.

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