# Melnikov’s Criteria and Chaos Analysis in the Nonlinear Schrödinger Equation with Kerr Law Nonlinearity

Authors: Jiuli Yin; Liuwei Zhao; Lixin Tian
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

The dynamics of the nonlinear Schrödinger equation with Kerr law nonlinearity with two perturbation terms are investigated. By using Melnikov method, the threshold values of chaotic motion under periodic perturbation are given. Moreover we also study the effects of the parameters of system on dynamical behaviors by using numerical simulation. The numerical simulations, including bifurcation diagram of fixed points, chaos threshold diagram of system in three-dimensional space, maximum Lyapunov exponent,  and phase portraits, are also plotted to illustrate theoretical analysis and to expose the complex dynamical behaviors. In particular, we observe that the system can leave chaotic region to periodic motion by adjusting controllere, amplitude d
1, and frequency ω
2 of external forcing which can be considered a control strategy, and when the frequenciesy ω
2 and ω
1 approach the maximum frequency of disturbance, the system turmoil intensifies and control intensity increases.


---

## Body

## 1. Introduction

Fiber-optic signal transmission has a very wide application in the real life, which makes our lives more convenient and quicker. This is known to all. The fiber-optic signal plays an important role in real life. Optical solitary waves have been the subject of intense current research, which is motivated by their important applications in the areas of high-capacity fiber telecommunications and all-optical switches due to their capability of propagating over a long distance without attenuation and changing their shapes [1].As is well known, the fiber-optic signal plays an important role in real life. It is the optical soliton propagation to maintain shape, amplitude and velocity constant light pulses for a long time. The use of optical solitons can achieve ultra-long-distance, large-capacity optical communications.Fiber-optic signals make it seem that the signal propagation cannot exist in pure environment. It is always influenced by external environmental perturbations. It seems that chaos may be unavoidable with external perturbations and has been observed in many practical applications such as engineering, biology, industry, and production. Besides, many other systems with external perturbations have been widely investigated by using analytic methods and numerical simulations [2–5].However, there is little to be subject to external disturbances as a model of optical signals. In fact, the fiber-optic signal propagation will be affected by lots of interferences in the process of real propagation. Moreover, the dynamic characteristics of nonlinear system in two-frequency interference will be richer and more complicated. We therefore think it is worthwhile to undertake a detailed discussion for system (1) in order to point out which range of parameter corresponds to certain behavior. Consequently, the two-frequency disturbance has more practical meaning.One of the famous fiber-optic models is the external periodic perturbations of the nonlinear Schrödinger equation with Kerr law nonlinearity [6]
(1)
i
u
t
+
u
x
x
+
α
|
u
|
2
u
+
i
[
γ
1
u
x
x
x
+
γ
2
|
u
|
2
u
x
+
γ
3
(
|
u
|
2
)
x
u
]
=
f
(
x
,
t
)
,
where γ
1 is the third-order dispersion and γ
2 and γ
3 are the nonlinear dispersion. Here γ
i   (i
=
1,2
,
3) are real parameters and α is a positive constant, f
(
x
,
t
)
=
e
-
i
Ω
t
(
d
1
cos
⁡
ω
1
(
x
-
c
t
)
+
d
2
cos
⁡
ω
2
(
x
-
c
t
)
), where the new variables d
i, w
i (i
=
1,2) denote the amplitude and the frequency of the parametric excitation, respectively. Here d
i and w
i (i
=
1,2) are real parameters. For this reason, d
i, w
i
≥
0  (i
=
1,2).Equation (1) describes the propagation of optical solitons in nonlinear optical fibers that exhibits a Kerr law nonlinearity. Equation (1) has an important application in various fields such as semiconductor materials, optical fiber communications, plasma physics, fluid and solid mechanics [6–12], and many other applications, can readily be described by the model, or analogous ones. Equation (1) has received more attention [6–10, 13, 14]. For example, by the first integral method, Taghizadeh et al. obtained the new exact solutions of the perturbed NLSE with Kerr law nonlinearity [6]. Wiggins [12] used the (G
′
/
G)-expansion method to construct traveling wave solutions for NLSEs. Masemola et al. [9] obtained new solutions of nonlinear partial differential equation by a direct algebraic method. Miao and Zhang [11] constructed exact solutions of perturbed nonlinear Schrödinger’s equation with Kerr law nonlinearity by the simplest equation method.The study of perturbed nonlinear Schrödinger’s equation with Kerr law nonlinearity is of fundamental and even practical interest. Eminent characteristics of perturbed nonlinear Schrödinger’s equation with Kerr law nonlinearity have a rich content of nonlinear properties which are suitable for a detailed investigating of various dynamical states.In this paper, we will study the chaotic behavior and control of (1) under two-frequency perturbations. We study the bifurcation and dynamical behaviors depending on the parameters by using bifurcation and chaos theories in [15–21] and numerical simulation. By applying Melnikov method [15, 20, 21], we prove the criterion of existence of chaos under periodic perturbation. Our interests have the following two points.(i)
What if there appears a system chaos under the two-frequency interference? If chaos appears, how does one design a controller to suppress chaos owing to the complex nonlinear item of system? Compared with the Duffing system [4, 5, 12], it is easy to see that there is no damping in our system. Once perturbed with external forcing, the system may easily tend to the chaotic state. Therefore, we will select the controller which has the same function with the damping and we consider the following equation:
(2)
i
u
t
+
u
x
x
+
α
|
u
|
2
u
+
i
[
γ
1
u
x
x
x
+
γ
2
|
u
|
2
u
x
+
γ
3
(
|
u
|
2
)
x
u
]
=
f
(
x
,
t
)
-
e
u
x
,
where e denotes the controller’s strength.
(ii)
We also study the effects of the parameters of system on dynamical behaviors by using numerical simulation. The numerical simulations, including bifurcation diagram of fixed points, chaos threshold diagram of system in three-dimensional space, maximum Lyapunov exponent, and phase portraits, are also plotted to illustrate theoretical analysis and to expose the complex dynamical behaviors. In particular, we observe that the system can leave chaotic region to periodic motion by adjusting controllere, amplitude d
1, and frequency ω
2 of external forcing which can be considered a control strategy.This paper is organized as follows. In Section2, we briefly describe the fixed points and phase portraits for the two types of unperturbed system of (1). In Section 3, the conditions of existence of chaos under periodic perturbation resulting from the homoclinic bifurcations are performed. The numerical investigations are given in Section 4. The bifurcation analysis is given in Section 5. Finally, we give remark to conclude this paper in Section 6.


## 2. Theoretic Analysis of System

### 2.1. The NLSE with Kerr Law Nonlinearity Equation

Assume that (1) has traveling wave solutions in the form [6]
(3)
u
(
x
,
t
)
=
ϕ
(
ξ
)
exp
⁡
(
-
i
Ω
t
)
,
ξ
=
x
-
c
t
,
where c is the propagation speed of a wave (c
>
0).Substituting (3) into (1), we get
(4)
i
(
γ
1
ϕ
′′′
+
γ
2
ϕ
2
ϕ
′
+
2
γ
2
ϕ
2
ϕ
′
-
c
ϕ
′
)
+
(
Ω
ϕ
+
ϕ
′′
+
α
ϕ
3
)
=
d
1
cos
⁡
(
w
1
ξ
)
+
d
2
cos
⁡
(
w
2
ξ
)
,
where γ
i   (i
=
1,2
,
3), α, and k are positive constants and prime meaning differentiation with respect to ξ.By virtue of [6], we have
(5)
γ
1
ϕ
′′
-
c
ϕ
+
A
ϕ
3
=
d
1
cos
⁡
(
w
1
ξ
)
+
d
2
cos
⁡
(
w
2
ξ
)
,
where A
=
(
1
/
3
)
γ
2
+
(
2
/
3
)
γ
3 denote the parametric of linear and nonlinear terms. Equation (5) is the fiber-optic signal transmission system in ideal environment. But it seems that the signal propagation cannot exist in pure environment. It is always influenced by external environmental perturbations.Without loss of generality in (5), we set γ
1
=
1. By the transformations ϕ
=
x
1, ϕ
′
=
x
2, (6) can be transformed into first-order nonautonomous equation:
(6)
x
1
′
=
x
2
,
x
2
′
=
c
x
1
-
A
x
1
3
+
d
1
cos
⁡
(
w
1
ξ
)
+
d
2
cos
⁡
(
w
2
ξ
)
.


### 2.2. Fixed Points and Phase Portraits for Unperturbed System

Ifd
1
=
d
2
=
0, (6) is considered an unperturbed system and can be written as
(7)
x
1
′
=
x
2
,
x
2
′
=
c
x
1
-
A
x
1
3
.The system (7) is a Hamiltonian system with Hamiltonian function
(8)
H
(
x
1
,
x
2
)
=
1
2
x
2
2
-
1
2
c
x
1
2
+
1
4
A
x
1
4
.And the function(9)
V
(
x
1
)
=
-
1
2
c
x
1
2
+
1
4
A
x
1
4
is called the potential function. By the analysis of the fixed points (x
1
j
,
x
2
j) and their stabilities for system (7), we can easily obtain the following results.Lemma 1.
Whenc
A
>
0, the unperturbed system has three equilibrium points: two centers (
c
/
A
,
0
) and (
-
c
/
A
,
0
) and one saddle (
0,0
). The saddle is connected to itself by two symmetric homoclinic orbits as shown in Figure 1(b).(a) The two-well potential function of the system; (b) the corresponding phase space portraits.
(a)
(b)


## 2.1. The NLSE with Kerr Law Nonlinearity Equation

Assume that (1) has traveling wave solutions in the form [6]
(3)
u
(
x
,
t
)
=
ϕ
(
ξ
)
exp
⁡
(
-
i
Ω
t
)
,
ξ
=
x
-
c
t
,
where c is the propagation speed of a wave (c
>
0).Substituting (3) into (1), we get
(4)
i
(
γ
1
ϕ
′′′
+
γ
2
ϕ
2
ϕ
′
+
2
γ
2
ϕ
2
ϕ
′
-
c
ϕ
′
)
+
(
Ω
ϕ
+
ϕ
′′
+
α
ϕ
3
)
=
d
1
cos
⁡
(
w
1
ξ
)
+
d
2
cos
⁡
(
w
2
ξ
)
,
where γ
i   (i
=
1,2
,
3), α, and k are positive constants and prime meaning differentiation with respect to ξ.By virtue of [6], we have
(5)
γ
1
ϕ
′′
-
c
ϕ
+
A
ϕ
3
=
d
1
cos
⁡
(
w
1
ξ
)
+
d
2
cos
⁡
(
w
2
ξ
)
,
where A
=
(
1
/
3
)
γ
2
+
(
2
/
3
)
γ
3 denote the parametric of linear and nonlinear terms. Equation (5) is the fiber-optic signal transmission system in ideal environment. But it seems that the signal propagation cannot exist in pure environment. It is always influenced by external environmental perturbations.Without loss of generality in (5), we set γ
1
=
1. By the transformations ϕ
=
x
1, ϕ
′
=
x
2, (6) can be transformed into first-order nonautonomous equation:
(6)
x
1
′
=
x
2
,
x
2
′
=
c
x
1
-
A
x
1
3
+
d
1
cos
⁡
(
w
1
ξ
)
+
d
2
cos
⁡
(
w
2
ξ
)
.


## 2.2. Fixed Points and Phase Portraits for Unperturbed System

Ifd
1
=
d
2
=
0, (6) is considered an unperturbed system and can be written as
(7)
x
1
′
=
x
2
,
x
2
′
=
c
x
1
-
A
x
1
3
.The system (7) is a Hamiltonian system with Hamiltonian function
(8)
H
(
x
1
,
x
2
)
=
1
2
x
2
2
-
1
2
c
x
1
2
+
1
4
A
x
1
4
.And the function(9)
V
(
x
1
)
=
-
1
2
c
x
1
2
+
1
4
A
x
1
4
is called the potential function. By the analysis of the fixed points (x
1
j
,
x
2
j) and their stabilities for system (7), we can easily obtain the following results.Lemma 1.
Whenc
A
>
0, the unperturbed system has three equilibrium points: two centers (
c
/
A
,
0
) and (
-
c
/
A
,
0
) and one saddle (
0,0
). The saddle is connected to itself by two symmetric homoclinic orbits as shown in Figure 1(b).(a) The two-well potential function of the system; (b) the corresponding phase space portraits.
(a)
(b)


## 3. Melnikov Theoretic Analysis

In this section, we discuss the chaotic behaviors of (6). Melnikov theory has proved to be a simple, elegant, and successful alternative to characterizing the complex dynamics of multistable oscillators. This section, thus, develops a global analysis technique, known as Melnikov’s method, to find the necessary conditions for homoclinic bifurcation to occur. For a detailed derivation of Melnikov’s method, there are several texts of varying rigor and sophistication to which the reader is referred [12, 22–24].In this section, we discuss the chaotic behaviors of (6) in which d
1 and d
2 are assumed to be small parameters. Transformations of d
1
→
ε
d
1 and d
2
→
ε
d
2 are done in order to apply the ε first-order perturbation scheme of the Melnikov theory. Hence, the system of (6) may be written as
(10)
x
1
′
=
x
2
,
x
2
′
=
c
x
1
-
A
x
1
3
+
ε
d
1
cos
⁡
(
ϕ
1
)
+
ε
d
2
cos
⁡
(
ϕ
2
)
,
ϕ
1
′
=
ω
1
,
ϕ
2
′
=
ω
2
.


### 3.1. Melnikov Criterion for Chaos

The unperturbed system for system (7) has homoclinic orbits. When the perturbation is added, the closed homoclinic orbits break and may have transverse homoclinic or heteroclinic orbits. By the Smale-Birkhoff Theorem [12, 25], the existence of such orbits results in chaotic dynamics. We therefore apply the Melnikov method to system (7) for finding the criteria of the existence of homoclinic or heteroclinic bifurcation and chaos.The Melnikov method derives a function to describe the first order distance between perturbed stable and unperturbed manifolds. Suppose that the unperturbed homoclinic or heteroclinic orbit is written as(
x
1
,
x
2
)
=
(
x
1
(
ξ
)
,
x
2
(
ξ
)
). Satisfying the conditions for a double-well potential gives rise to a homoclinic orbit in the phase space for ε
=
0. The homoclinic trajectory can be found by setting H
(
a
,
b
)
=
0. Solving for the resulting displacement and differentiating to determine velocity, the homoclinic trajectory is given as follows:
(11)
(
x
1
,
x
2
)
=
(
2
c
A
sech
(
c
ξ
)
,
-
2
c
A
c
sech
(
c
ξ
)
tanh
(
c
ξ
)
)
,
and then the Melnikov function for system (7) can be given by
(12)
M
(
ξ
0
)
=
∫
-
∞
∞
x
2
(
ξ
)
[
d
1
cos
⁡
w
1
(
ξ
+
ξ
0
)
+
d
2
cos
⁡
w
2
(
ξ
+
ξ
0
)
]
d
ξ
,
where ξ
0 is the cross-section time of the Poincaré map and ξ
0 can be interpreted as the initial time of the forcing term.Because it is difficult to give analytical expression ofx
2
(
ξ
), we will compute x
2
(
ξ
) numerically in Section 5. We note that x
2
(
ξ
) is a function of time from +
∞ to -
∞. We therefore choose that the initial condition is ξ
=
0 and x
2
(
ξ
) would be an odd function of time for the homoclinic orbit and an even function for heteroclinic orbit.For the homoclinic orbitsΓ
ho
±, the Melnikov function can be simplified as
(13)
M
1
(
ξ
0
)
=
∫
-
∞
∞
x
2
(
ξ
)
[
d
1
cos
⁡
w
1
(
ξ
+
ξ
0
)
+
d
2
cos
⁡
w
2
(
ξ
+
ξ
0
)
]
d
ξ
=
-
2
d
1
sin
w
1
ξ
0
∫
0
∞
x
2
(
ξ
)
sin
w
1
ξ
d
ξ
-
2
d
2
sin
w
2
ξ
0
∫
0
∞
x
2
(
ξ
)
sin
w
2
ξ
d
ξ
=
-
2
d
1
sin
w
1
ξ
0
I
hom
1
(
w
1
)
-
2
d
2
sin
w
2
ξ
0
I
hom
2
(
w
2
)
,
where I
hom
1
=
∫
0
∞
x
2
(
ξ
)
sin
w
1
ξ
d
ξ and I
hom
2
=
∫
0
∞
x
2
(
ξ
)
sin
w
2
ξ
d
ξ are functions of the frequencies w
1 and w
2, respectively.Using the previous results and Melnikov’s theorem [26, 27], the following is stated: if M
1
(
ξ
0
)
=
0 and M
1
′
(
ξ
0
)
≠
0 for some ξ
0 and some set of parameters, then horseshoes exist and chaos occurs [12, 25]. If M
1
(
ξ
0
) has a simple zero and the corresponding critical parameter value is
(14)
(
e
)
0
=
|
d
1
I
hom
1
(
ω
1
)
+
d
2
I
hom
2
(
ω
2
)
|
B
,
where B
=
∫
0
∞
x
2
2
(
ξ
)
d
ξ is a constant once x
2
(
ξ
) is given, then in the system with fractional order displacement (11) the deterministic chaos may appear for certain parameter values which satisfy the relation
(15)
e
<
(
e
)
0
.Remark 2.
Using the Melnikov criterion for the appearance of the intersection between the perturbed and unperturbed separatrices, therefore, for fixed frequenciesω
1 and ω
2, the system always produces Smale commutation of chaos.


### 3.2. Control of Chaos

Because the fiber-optic transmission system in the chaotic state is very sensitive to its initial condition and chaos often causes irregular behavior, chaos is undesirable. It is not hard to see that system (6) is similar to Duffing system, except the absence of damping in the former. Therefore, we will select the controller that has the same function with the damping.Equation (2) is equivalent to the following system:
(16)
-
c
x
1
+
x
1
′′
+
A
x
1
3
=
d
1
cos
⁡
(
ω
1
ξ
)
+
d
2
cos
⁡
(
ω
2
ξ
)
-
e
x
1
′
.Equation (16) can be transformed into first-order nonautonomous equation:
(17)
x
1
′
=
x
2
,
x
2
′
=
c
x
1
-
A
x
1
3
+
ε
d
1
cos
⁡
(
w
1
ξ
)
+
ε
d
2
cos
⁡
(
w
2
ξ
)
-
ε
e
x
2
,
where d
1
→
ε
d
1, d
2
→
ε
d
2, and e
→
ε
e.Now, the Melnikov function for system (17) can be given by
(18)
M
1
(
ξ
0
)
=
∫
-
∞
∞
x
2
(
ξ
)
[
-
e
x
2
(
ξ
)
+
d
1
cos
⁡
ω
1
(
ξ
+
ξ
0
)
+
d
2
cos
⁡
ω
2
(
ξ
+
ξ
0
)
]
d
ξ
,
and the Melnikov function can be simplified as
(19)
M
1
(
ξ
0
)
=
-
2
e
∫
0
∞
x
2
2
(
ξ
)
d
ξ
-
2
d
1
sin
w
1
ξ
0
∫
0
∞
x
2
(
ξ
)
sin
w
1
ξ
d
ξ
-
2
d
2
sin
w
2
ξ
0
∫
0
∞
x
2
(
ξ
)
sin
w
2
ξ
d
ξ
=
-
2
e
B
-
2
d
1
sin
w
1
ξ
0
I
hom
1
(
w
1
)
-
2
d
2
sin
w
2
ξ
0
I
hom
2
(
w
2
)
.Thus, if(20)
e
<
|
d
1
I
hom
1
(
ω
1
)
+
d
2
I
hom
2
(
ω
2
)
|
B
≡
R
(
ω
)
,
then there is a ξ
0
¯ such that M
1
(
ξ
0
¯
) and (
∂
M
1
/
∂
ξ
0
)
|
ξ
=
ξ
0
≠
0, ξ
0
¯
∈
[
0
,
2
π
/
ω
1
], and the following lemma can be obtained.Lemma 3.
The homoclinic bifurcation will occur at(21)
e
=
R
(
ω
)
.Remark 4.
For (20) the distance between the stable and unstable manifolds of the homoclinic point (0,0) is zero and the manifolds intersect transversely forming the transverse homoclinic orbits. The presence of such orbits implies that, for certain parameters (21) and the countable infinity of unstable periodic orbits, an uncountable set of bounded nonperiodic orbits and a dense orbit are the main characteristics of the chaotic motion.


## 3.1. Melnikov Criterion for Chaos

The unperturbed system for system (7) has homoclinic orbits. When the perturbation is added, the closed homoclinic orbits break and may have transverse homoclinic or heteroclinic orbits. By the Smale-Birkhoff Theorem [12, 25], the existence of such orbits results in chaotic dynamics. We therefore apply the Melnikov method to system (7) for finding the criteria of the existence of homoclinic or heteroclinic bifurcation and chaos.The Melnikov method derives a function to describe the first order distance between perturbed stable and unperturbed manifolds. Suppose that the unperturbed homoclinic or heteroclinic orbit is written as(
x
1
,
x
2
)
=
(
x
1
(
ξ
)
,
x
2
(
ξ
)
). Satisfying the conditions for a double-well potential gives rise to a homoclinic orbit in the phase space for ε
=
0. The homoclinic trajectory can be found by setting H
(
a
,
b
)
=
0. Solving for the resulting displacement and differentiating to determine velocity, the homoclinic trajectory is given as follows:
(11)
(
x
1
,
x
2
)
=
(
2
c
A
sech
(
c
ξ
)
,
-
2
c
A
c
sech
(
c
ξ
)
tanh
(
c
ξ
)
)
,
and then the Melnikov function for system (7) can be given by
(12)
M
(
ξ
0
)
=
∫
-
∞
∞
x
2
(
ξ
)
[
d
1
cos
⁡
w
1
(
ξ
+
ξ
0
)
+
d
2
cos
⁡
w
2
(
ξ
+
ξ
0
)
]
d
ξ
,
where ξ
0 is the cross-section time of the Poincaré map and ξ
0 can be interpreted as the initial time of the forcing term.Because it is difficult to give analytical expression ofx
2
(
ξ
), we will compute x
2
(
ξ
) numerically in Section 5. We note that x
2
(
ξ
) is a function of time from +
∞ to -
∞. We therefore choose that the initial condition is ξ
=
0 and x
2
(
ξ
) would be an odd function of time for the homoclinic orbit and an even function for heteroclinic orbit.For the homoclinic orbitsΓ
ho
±, the Melnikov function can be simplified as
(13)
M
1
(
ξ
0
)
=
∫
-
∞
∞
x
2
(
ξ
)
[
d
1
cos
⁡
w
1
(
ξ
+
ξ
0
)
+
d
2
cos
⁡
w
2
(
ξ
+
ξ
0
)
]
d
ξ
=
-
2
d
1
sin
w
1
ξ
0
∫
0
∞
x
2
(
ξ
)
sin
w
1
ξ
d
ξ
-
2
d
2
sin
w
2
ξ
0
∫
0
∞
x
2
(
ξ
)
sin
w
2
ξ
d
ξ
=
-
2
d
1
sin
w
1
ξ
0
I
hom
1
(
w
1
)
-
2
d
2
sin
w
2
ξ
0
I
hom
2
(
w
2
)
,
where I
hom
1
=
∫
0
∞
x
2
(
ξ
)
sin
w
1
ξ
d
ξ and I
hom
2
=
∫
0
∞
x
2
(
ξ
)
sin
w
2
ξ
d
ξ are functions of the frequencies w
1 and w
2, respectively.Using the previous results and Melnikov’s theorem [26, 27], the following is stated: if M
1
(
ξ
0
)
=
0 and M
1
′
(
ξ
0
)
≠
0 for some ξ
0 and some set of parameters, then horseshoes exist and chaos occurs [12, 25]. If M
1
(
ξ
0
) has a simple zero and the corresponding critical parameter value is
(14)
(
e
)
0
=
|
d
1
I
hom
1
(
ω
1
)
+
d
2
I
hom
2
(
ω
2
)
|
B
,
where B
=
∫
0
∞
x
2
2
(
ξ
)
d
ξ is a constant once x
2
(
ξ
) is given, then in the system with fractional order displacement (11) the deterministic chaos may appear for certain parameter values which satisfy the relation
(15)
e
<
(
e
)
0
.Remark 2.
Using the Melnikov criterion for the appearance of the intersection between the perturbed and unperturbed separatrices, therefore, for fixed frequenciesω
1 and ω
2, the system always produces Smale commutation of chaos.


## 3.2. Control of Chaos

Because the fiber-optic transmission system in the chaotic state is very sensitive to its initial condition and chaos often causes irregular behavior, chaos is undesirable. It is not hard to see that system (6) is similar to Duffing system, except the absence of damping in the former. Therefore, we will select the controller that has the same function with the damping.Equation (2) is equivalent to the following system:
(16)
-
c
x
1
+
x
1
′′
+
A
x
1
3
=
d
1
cos
⁡
(
ω
1
ξ
)
+
d
2
cos
⁡
(
ω
2
ξ
)
-
e
x
1
′
.Equation (16) can be transformed into first-order nonautonomous equation:
(17)
x
1
′
=
x
2
,
x
2
′
=
c
x
1
-
A
x
1
3
+
ε
d
1
cos
⁡
(
w
1
ξ
)
+
ε
d
2
cos
⁡
(
w
2
ξ
)
-
ε
e
x
2
,
where d
1
→
ε
d
1, d
2
→
ε
d
2, and e
→
ε
e.Now, the Melnikov function for system (17) can be given by
(18)
M
1
(
ξ
0
)
=
∫
-
∞
∞
x
2
(
ξ
)
[
-
e
x
2
(
ξ
)
+
d
1
cos
⁡
ω
1
(
ξ
+
ξ
0
)
+
d
2
cos
⁡
ω
2
(
ξ
+
ξ
0
)
]
d
ξ
,
and the Melnikov function can be simplified as
(19)
M
1
(
ξ
0
)
=
-
2
e
∫
0
∞
x
2
2
(
ξ
)
d
ξ
-
2
d
1
sin
w
1
ξ
0
∫
0
∞
x
2
(
ξ
)
sin
w
1
ξ
d
ξ
-
2
d
2
sin
w
2
ξ
0
∫
0
∞
x
2
(
ξ
)
sin
w
2
ξ
d
ξ
=
-
2
e
B
-
2
d
1
sin
w
1
ξ
0
I
hom
1
(
w
1
)
-
2
d
2
sin
w
2
ξ
0
I
hom
2
(
w
2
)
.Thus, if(20)
e
<
|
d
1
I
hom
1
(
ω
1
)
+
d
2
I
hom
2
(
ω
2
)
|
B
≡
R
(
ω
)
,
then there is a ξ
0
¯ such that M
1
(
ξ
0
¯
) and (
∂
M
1
/
∂
ξ
0
)
|
ξ
=
ξ
0
≠
0, ξ
0
¯
∈
[
0
,
2
π
/
ω
1
], and the following lemma can be obtained.Lemma 3.
The homoclinic bifurcation will occur at(21)
e
=
R
(
ω
)
.Remark 4.
For (20) the distance between the stable and unstable manifolds of the homoclinic point (0,0) is zero and the manifolds intersect transversely forming the transverse homoclinic orbits. The presence of such orbits implies that, for certain parameters (21) and the countable infinity of unstable periodic orbits, an uncountable set of bounded nonperiodic orbits and a dense orbit are the main characteristics of the chaotic motion.


## 4. Numerical Simulations

In this section, we give numerical simulations to support the theoretical results obtained in the previous sections and to find other new dynamics.The interesting problem is to analyze the parameter regions for optical fiber signals’ stable propagation of the controlled system. The controlled fiber-optic transmission system has several parameters and each of them plays different and virtual roles in the system. We will analyze the influence on optic-fiber signals propagation of controlled system (21) when the parameter of system changes with the fixed controller.Let us study the intersections of the invariant manifolds of the saddle point. It is known that these intersections are the necessary conditions for the existence of chaos. Since the Melnikov function theory measures the distance between the perturbed stable manifold and unstable manifolds, a homoclinic tangency will occur when a real solution can be found for some timeξ
0 such that the function has a simple zero. This means that only necessary conditions for the appearance of strange attractors are obtained from Poincaré-Melnikov-Arnold analysis, and therefore one always has the chance of finding sufficient conditions for the elimination of even transient chaos. Then the general necessary condition for which the invariant manifolds intersect is given by
(22)
e
=
R
(
ω
)
.We give numerical simulations to support the theoretical results as shown in Figures2–6.Figure 2
Chaotic threshold in(
d
1
,
w
1
,
e
) space.Figure 3
Chaotic threshold in(
ω
1
,
A
,
e
) space.Figure 4
Chaotic threshold in(
ω
2
,
d
1
,
e
) space.Figure 5
Chaotic threshold in(
ω
1
,
ω
2
,
e
) space.Figure 6
Chaotic threshold in(
d
1
,
d
2
,
e
) space.The chaotic threshold in(
d
1
,
ω
1
,
e
) space with d
2
=
1, ω
2
=
0.08, A
=
1, and c
=
0.5 is shown in Figure 2. The chaotic threshold in (
A
,
ω
1
,
e
) space with d
2
=
1, ω
2
=
0.08, d
1
=
1.2, and c
=
0.5 is shown in Figure 3. The chaotic threshold in (
ω
2
,
d
1
,
e
) space with d
2
=
1, ω
1
=
0.05, A
=
1.2, and c
=
1 is shown in Figure 4. The chaotic threshold in (
ω
2
,
ω
1
,
e
) space with d
2
=
1, d
1
=
1.2, A
=
0.5, and c
=
0.8 is shown in Figure 5. The chaotic threshold in (
d
2
,
d
1
,
e
) space with A
=
0.5, ω
2
=
1, ω
1
=
1.2, and c
=
0.8 is shown in Figure 6. When the value of e is below the surface, the system may be chaotic state. In order to show what happens to the solutions and attractors as one crosses these bifurcation surfaces, we chose the parameter values from the regions e
<
(
e
)
0, and then the system (17) can exhibit chaos.Remark 5.
This implies that ifε
>
0 is sufficiently small, the reduced equation (17) has transverse homoclinic orbits resulting in possible chaotic dynamic. With c and A constant, we study chaotic threshold as a function of only the frequency parameter ω. A typical plot of e is shown in Figure 2. The qualitative form of this function remains the same as c and A are the values for which the potential is two-well. Another point is that, when the ratio e tends to zero, this means that the external amplitude d
1 tends to infinity and then the Melnikov theory is not valid for these values.


## 5. Bifurcation Analysis

In this section, we give numerical simulations to support the theoretical results of control of chaos.


### 5.1. Perturbation and Controlled System Analysis

We will discuss the behaviors of the fiber-optic signal transmission under perturbation and we draw the bifurcation diagram of (6) in (d
1
,
x
2) plane and the corresponding maximal Lyapunov exponents in Figure 7. System (6) is integrated using the Runge-Kutta technique of order four to conduct numerical simulation. Numerical calculations have been made for the selected parameter values A
=
0.2, c
=
0.0001, d
2
=
1.5, ω
1
=
1.2, and ω
2
=
1 with the initial conditions x
1
=
1 and x
2
=
0. Since the equation is nonlinear, its solution therefore admits the possibility of periodic and chaotic orbits.(a) Bifurcation diagram of system (6) in (d
1
,
x
2) plane 0
≤
d
≤
4. (b) Maximum Lyapunov exponents corresponding to (a).
(a)
(b)In Figure7, we can observe that the value of Lyapunov exponents is positive (a positive top Lyapunov exponent for a bounded attractor is usually a sign of chaos), so the system easily converts to chaos even if there is small perturbation. It indicates that the system is outside perturbed, as the optical fiber transmission signal is very vulnerable to the phenomenon of chaos. So an appropriate controller is needed to satisfy the practical applications of fiber-optic propagation. Such maps can be used to suppress chaotic dynamics of the system.Next the corresponding numerical simulations are performed for the case of periodic perturbation of system (16); we select d
=
1. In Figure 8(a) after a large band of chaotic regime for e
∈
(
0,0.045
) one can find a sequence of backward period-doubling bifurcations as a route to periodic motion after a periodic regime; another bifurcation takes place at a critical value e
=
0.062 where another large band of chaotic regime with small periodic window occurs. At e
∈
(
0.046,0.061
) and e
>
0.063, the system displays periodic behavior after backward period-doubling bifurcations. Such maps can be used to suppress chaotic dynamics of the system.(a) Bifurcation diagram of system (17) in (e
,
x
2) plane 0
≤
e
≤
4. (b) Maximum Lyapunov exponents corresponding to (a). (c) Enlarged view of (b).
(a)
(b)
(c)From Figure8(b) one can see that for smaller values of controller e, the top Lyapunov exponent is positive. As e increases, the top Lyapunov exponent changes from positive value to negative value, signifying the suppressing of homoclinic chaos motion. Beyond the threshold for onset of chaotic motion, there are some “periodic windows;” this can be the feature of the transient chaos. However, the larger control parameters are responsive for the decreasing signal amplitude. It is easy to see that the signal cannot propagate normally and might leak from the media, which is called escape.Remark 6.
According to the above analysis, the increasing of the controller’s coefficiente makes the system stable, but escape occurs when e crosses a certain value.


### 5.2. Parameters Analysis

Now we wish to find other interesting bifurcation structures and dynamics of system (17). The bifurcation parameters are considered in the following five cases:(i)
varyingd
1 in range 0
<
d
1
<
4 and fixing A
=
0.2, c
=
0.0001, e
=
0.1, ω
1
=
1.2, and d
2
=
1.5 for several values of ω
2,
(ii)
varyingd
2 in range 0
<
d
2
<
4 and fixing A
=
0.2, c
=
0.0001, e
=
0.1, ω
1
=
1.2, and d
1
=
1 for several values of ω
2,
(iii)
varyingω
2 in range 0
<
ω
2
<
4 and fixing A
=
0.2, c
=
0.0001, ω
1
=
1, d
1
=
0.5, and d
2
=
0.5 for several values of e,
(iv)
varyinge in range 0
<
e
<
0.4 and fixing ω
2
=
2, d
2
=
0.5, ω
1
=
1.2, d
1
=
1.8, and c
=
0.0002 for several values of A,
(v)
varyinge in range 0
<
e
<
0.4 and fixing A
=
0.2, d
1
=
1, ω
1
=
1.2, d
2
=
1.2, and ω
2
=
2 for several values of c.For cases (i) and (ii), we give bifurcation diagrams in (d
1
,
x
2) and (d
2
,
x
2) plane of system (17) for 0
<
d
1
<
4 and 0
<
d
2
<
4 and several values of ω
2 in Figures 9 and 10, respectively. We observe that a wide chaotic region in Figure 9(a). The coexistences of periodic orbits and chaotic motions in chaotic windows and intermittent mechanism in Figure 9(b). Figure 9(c) shows that the onset of chaos after period-n and chaotic regions with complex periodic windows, Figure 9(d) also shows the onset of chaos and chaotic regions with small periodic windows. Comparing Figures 10(a)–10(d), we find that, as ω
2 is increased, the complexity of dynamical behaviors is decreased, and as ω
2
=
4 there is only a period-one orbit and there is no bifurcation (see Figure 10(d)).Bifurcation diagrams of system (17) in (d
1
,
x
2) plane for case (iii): (a) ω
2
=
0.8, (b) ω
2
=
1.8, (c) ω
2
=
2.5, and (d) ω
2
=
3.
(a)
(b)
(c)
(d)Bifurcation diagrams of system (17) in (d
2
,
x
2) plane for case (iii): (a) ω
2
=
0.7, (b) ω
2
=
1.5, (c) ω
2
=
3.2, and (d) ω
2
=
4.
(a)
(b)
(c)
(d)For case (iii), the bifurcation diagrams in (ω
2
,
x
2) plane and the corresponding maximum Lyapunov exponents are given in Figure 11, where we show that the maximum Lyapunov exponents are all negative in Figure 11(d) for e
=
0.25 which are corresponding to quasiperiodic solutions in Figure 11(b). Figures 11(a) and 11(c) exhibit the process of the quasiperiodic route to chaos and chaos converges to quasiperiodic orbit.Bifurcation diagrams of system (17) in (ω
2
,
x
2) plane for case (iii): (a) e
=
0.1; (b) e
=
0.25. The maximum Lyapunov exponents: (c) corresponding to (a), (d) corresponding to (b).
(a)
(b)
(c)
(d)For case (iv), the bifurcation diagrams in Figures12(a)–12(d) in (e
,
x
2) plane for various values of A are plotted for showing the effect of parameter A on dynamical behaviors. When A is increased a little, the chaotic behavior occurrence has changed and the amplitude had decreased a little in Figure 12(b). But, as A is further increased, there is only the period-five orbit and a period-doubling bifurcation of period-one.Bifurcation diagrams of system (17) in (e
,
x
2) plane for case (iii): (a) A
=
0.3, (b) A
=
0.5, (c) A
=
1, and (d) A
=
2.
(a)
(b)
(c)
(d)For case (v), the bifurcation diagrams in Figures13(a)–13(d) in (e
,
x
2) plane for various values of c are plotted for showing the effect of parameter c on dynamical behaviors. We show that the chaotic behavior occurs alternately in smaller region of c and the chaotic motions behavior occurs alternately in bigger region as a is increased. Therefore, we find that the parameter c plays a very important role for dynamical behaviors from comparing. It can be considered as an control strategy of chaos by adjusting the parameter e.Bifurcation diagrams of system (17) in (e
,
x
2) plane for case (iii): (a) c
=
0.002, (b) c
=
0.02, (c) c
=
0.8, and (d) c
=
1.2.
(a)
(b)
(c)
(d)


## 5.1. Perturbation and Controlled System Analysis

We will discuss the behaviors of the fiber-optic signal transmission under perturbation and we draw the bifurcation diagram of (6) in (d
1
,
x
2) plane and the corresponding maximal Lyapunov exponents in Figure 7. System (6) is integrated using the Runge-Kutta technique of order four to conduct numerical simulation. Numerical calculations have been made for the selected parameter values A
=
0.2, c
=
0.0001, d
2
=
1.5, ω
1
=
1.2, and ω
2
=
1 with the initial conditions x
1
=
1 and x
2
=
0. Since the equation is nonlinear, its solution therefore admits the possibility of periodic and chaotic orbits.(a) Bifurcation diagram of system (6) in (d
1
,
x
2) plane 0
≤
d
≤
4. (b) Maximum Lyapunov exponents corresponding to (a).
(a)
(b)In Figure7, we can observe that the value of Lyapunov exponents is positive (a positive top Lyapunov exponent for a bounded attractor is usually a sign of chaos), so the system easily converts to chaos even if there is small perturbation. It indicates that the system is outside perturbed, as the optical fiber transmission signal is very vulnerable to the phenomenon of chaos. So an appropriate controller is needed to satisfy the practical applications of fiber-optic propagation. Such maps can be used to suppress chaotic dynamics of the system.Next the corresponding numerical simulations are performed for the case of periodic perturbation of system (16); we select d
=
1. In Figure 8(a) after a large band of chaotic regime for e
∈
(
0,0.045
) one can find a sequence of backward period-doubling bifurcations as a route to periodic motion after a periodic regime; another bifurcation takes place at a critical value e
=
0.062 where another large band of chaotic regime with small periodic window occurs. At e
∈
(
0.046,0.061
) and e
>
0.063, the system displays periodic behavior after backward period-doubling bifurcations. Such maps can be used to suppress chaotic dynamics of the system.(a) Bifurcation diagram of system (17) in (e
,
x
2) plane 0
≤
e
≤
4. (b) Maximum Lyapunov exponents corresponding to (a). (c) Enlarged view of (b).
(a)
(b)
(c)From Figure8(b) one can see that for smaller values of controller e, the top Lyapunov exponent is positive. As e increases, the top Lyapunov exponent changes from positive value to negative value, signifying the suppressing of homoclinic chaos motion. Beyond the threshold for onset of chaotic motion, there are some “periodic windows;” this can be the feature of the transient chaos. However, the larger control parameters are responsive for the decreasing signal amplitude. It is easy to see that the signal cannot propagate normally and might leak from the media, which is called escape.Remark 6.
According to the above analysis, the increasing of the controller’s coefficiente makes the system stable, but escape occurs when e crosses a certain value.


## 5.2. Parameters Analysis

Now we wish to find other interesting bifurcation structures and dynamics of system (17). The bifurcation parameters are considered in the following five cases:(i)
varyingd
1 in range 0
<
d
1
<
4 and fixing A
=
0.2, c
=
0.0001, e
=
0.1, ω
1
=
1.2, and d
2
=
1.5 for several values of ω
2,
(ii)
varyingd
2 in range 0
<
d
2
<
4 and fixing A
=
0.2, c
=
0.0001, e
=
0.1, ω
1
=
1.2, and d
1
=
1 for several values of ω
2,
(iii)
varyingω
2 in range 0
<
ω
2
<
4 and fixing A
=
0.2, c
=
0.0001, ω
1
=
1, d
1
=
0.5, and d
2
=
0.5 for several values of e,
(iv)
varyinge in range 0
<
e
<
0.4 and fixing ω
2
=
2, d
2
=
0.5, ω
1
=
1.2, d
1
=
1.8, and c
=
0.0002 for several values of A,
(v)
varyinge in range 0
<
e
<
0.4 and fixing A
=
0.2, d
1
=
1, ω
1
=
1.2, d
2
=
1.2, and ω
2
=
2 for several values of c.For cases (i) and (ii), we give bifurcation diagrams in (d
1
,
x
2) and (d
2
,
x
2) plane of system (17) for 0
<
d
1
<
4 and 0
<
d
2
<
4 and several values of ω
2 in Figures 9 and 10, respectively. We observe that a wide chaotic region in Figure 9(a). The coexistences of periodic orbits and chaotic motions in chaotic windows and intermittent mechanism in Figure 9(b). Figure 9(c) shows that the onset of chaos after period-n and chaotic regions with complex periodic windows, Figure 9(d) also shows the onset of chaos and chaotic regions with small periodic windows. Comparing Figures 10(a)–10(d), we find that, as ω
2 is increased, the complexity of dynamical behaviors is decreased, and as ω
2
=
4 there is only a period-one orbit and there is no bifurcation (see Figure 10(d)).Bifurcation diagrams of system (17) in (d
1
,
x
2) plane for case (iii): (a) ω
2
=
0.8, (b) ω
2
=
1.8, (c) ω
2
=
2.5, and (d) ω
2
=
3.
(a)
(b)
(c)
(d)Bifurcation diagrams of system (17) in (d
2
,
x
2) plane for case (iii): (a) ω
2
=
0.7, (b) ω
2
=
1.5, (c) ω
2
=
3.2, and (d) ω
2
=
4.
(a)
(b)
(c)
(d)For case (iii), the bifurcation diagrams in (ω
2
,
x
2) plane and the corresponding maximum Lyapunov exponents are given in Figure 11, where we show that the maximum Lyapunov exponents are all negative in Figure 11(d) for e
=
0.25 which are corresponding to quasiperiodic solutions in Figure 11(b). Figures 11(a) and 11(c) exhibit the process of the quasiperiodic route to chaos and chaos converges to quasiperiodic orbit.Bifurcation diagrams of system (17) in (ω
2
,
x
2) plane for case (iii): (a) e
=
0.1; (b) e
=
0.25. The maximum Lyapunov exponents: (c) corresponding to (a), (d) corresponding to (b).
(a)
(b)
(c)
(d)For case (iv), the bifurcation diagrams in Figures12(a)–12(d) in (e
,
x
2) plane for various values of A are plotted for showing the effect of parameter A on dynamical behaviors. When A is increased a little, the chaotic behavior occurrence has changed and the amplitude had decreased a little in Figure 12(b). But, as A is further increased, there is only the period-five orbit and a period-doubling bifurcation of period-one.Bifurcation diagrams of system (17) in (e
,
x
2) plane for case (iii): (a) A
=
0.3, (b) A
=
0.5, (c) A
=
1, and (d) A
=
2.
(a)
(b)
(c)
(d)For case (v), the bifurcation diagrams in Figures13(a)–13(d) in (e
,
x
2) plane for various values of c are plotted for showing the effect of parameter c on dynamical behaviors. We show that the chaotic behavior occurs alternately in smaller region of c and the chaotic motions behavior occurs alternately in bigger region as a is increased. Therefore, we find that the parameter c plays a very important role for dynamical behaviors from comparing. It can be considered as an control strategy of chaos by adjusting the parameter e.Bifurcation diagrams of system (17) in (e
,
x
2) plane for case (iii): (a) c
=
0.002, (b) c
=
0.02, (c) c
=
0.8, and (d) c
=
1.2.
(a)
(b)
(c)
(d)


## 6. Conclusions

In this paper, we have investigated the behaviors of the nonlinear Schrödinger equation with Kerr law nonlinearity with two perturbation terms and find many complex and interesting dynamical behaviors by using analytic and numerical methods. We conclude that chaos occurs easily due to the absence of damping in the system. This phenomenon will cause the distortion in the process of information transmission. One can add a controller to suppress the chaos. The efficiency of this controller was demonstrated. The complex fiber-optic transmission system of the perturbed NLSE with Kerr law nonlinearity was controlled. What is more is that we discussed the sensitivity to be controlled and found the practical parameters regions. Some of the results obtained for other NLSEs can be extended correspondingly.



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*Source: 650781-2014-06-10.xml*