On Tzitéeica type nonlinear equations for multiple soliton solutions in nonlinear optics

Yongsheng Rao; Asim Zafar; Alper Korkmaz; Asfand Fahad; Muhammad Imran Qureshi; Yongsheng Rao; Asim Zafar; Alper Korkmaz; Asfand Fahad; Muhammad Imran Qureshi

doi:10.3934/math.2020423

AIMS Mathematics

2020, Volume 5, Issue 6: 6580-6593. doi: 10.3934/math.2020423

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Research article Special Issues

On Tzitéeica type nonlinear equations for multiple soliton solutions in nonlinear optics

1.
Institute of Computing Science and Technology, Guangzhou University, Guangzhou 511400, China
2.
Department of Mathematics, COMSATS University Islamabad, Vehari Campus, Vehari 61100, Pakistan
3.
The IJEMAPS Journal 99427 Weimar, Thuringen, Germany

Received: 05 July 2020 Accepted: 17 August 2020 Published: 24 August 2020
MSC : 35A09

Nonlinear equations personate a consequential role in scientific fields such as nonlinear optics, solid state physics and quantum field theory. This article studies the Tzitzéica-Dodd-Bullough-Mikhailov and Tzitzéica-type equations that appear in nonlinear optics. The Painlevé and traveling wave transformations both play a key role in revamping the aforementioned equations into nonlinear ordinary differential equations. Then, the simple ansatz approach is followed to seize complex singular, complex bright solitons and other kink type solutions. The existing literature unveils that this study is a novel contribution in the literature and the proposed approach is forthright and simple to implement for solving nonlinear problems. This approach has no extra condition as compared to many other techniques have. We also verify and interpret graphically the secured solutions through symbolic software Mathematica and MatLab respectively.

Keywords:

Citation: Yongsheng Rao, Asim Zafar, Alper Korkmaz, Asfand Fahad, Muhammad Imran Qureshi. On Tzitéeica type nonlinear equations for multiple soliton solutions in nonlinear optics[J]. AIMS Mathematics, 2020, 5(6): 6580-6593. doi: 10.3934/math.2020423

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Abstract

1. Introduction

Since pioneering works of Pecora and Carroll's ^[1], chaos synchronization and control have turned a hot topic and received much attention in various research areas. A number of literatures shows that chaos synchronization can be widely used in physics, medicine, biology, quantum neuron and engineering science, particularly in secure communication and telecommunications ^[1,2,3]. In order to realize synchronization, experts have proposed lots of methods, including complete synchronization and $Q$ - $S$ synchronization ^[4,5], adaptive synchronization ^[6], lag synchronization^[7,8], phase synchronization ^[9], observer-based synchronization ^[10], impulsive synchronization ^[11], generalized synchronization ^[12,13], lag projective synchronization ^[14,15], cascade synchronization et al ^{[16,17,18,19,20]}. Among them, the cascade synchronization method is a very effective algorithm, which is characterized by reproduction of signals in the original chaotic system to monitor the synchronized motions.

It is know that, because of the complexity of fractional differential equations, synchronization of fractional-order chaotic systems is more difficult but interesting than that of integer-order systems. Experts find that the key space can be enlarged by the regulating parameters in fractional-order chaotic systems, which enables the fractional-order chaotic system to be more suitable for the use of the encryption and control processing. Therefore, synchronization of fractional-order chaotic systems has gained increasing interests in recent decades ^{[21,22,23,24,25,26,27,28,29,30,31]}. It is noticed that most synchronization methods mentioned in ^{[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]} work for integer-order chaotic systems. Here, we shall extend to cascade synchronization for integer-order chaotic systems to a kind of general form, namely function cascade synchronization (FCS), which means that one chaotic system may be synchronized with another by sending a signal from one to the other wherein a scaling function is involved. The FCS is effective both for the fractional order and integer order chaotic systems. It constitutes a general method, which can be considered as a continuation and extension of earlier works of ^[13,16,19]. The nice feature of our method is that we introduce a scaling function for achieving synchronization of fractional-order chaotic systems, which can be chosen as a constant, trigonometric function, power function, logarithmic and exponential function, hyperbolic function and even combinations of them. Hence, our method is more general than some existing methods, such as the complete synchronization approach and anti-phase synchronization approach et al.

To sum up, in this paper, we would like to use the FCS approach proposed to study the synchronization of fractional-order chaotic systems. We begin our theoretical work with the Caputo fractional derivative. Then, we give the FCS of the fractional-order chaotic systems in theory. Subsequently, we take the fractional-order unified chaotic system as a concrete example to test the effectiveness of our method. Finally, we make a short conclusion.

2. The method of FCS for fractional-order chaotic systems

As for the fractional derivative, there exists a lot of mathematical definitions ^[32,33]. Here, we shall only adopt the Caputo fractional calculus, which allows the traditional initial and boundary condition assumptions. The Caputo fractional calculus is described by

$\begin{eqnarray} \frac{d^q f(t)}{dt^q} = \frac{1}{{\Gamma}(q-n)}\int^t_0 \frac{f^{(n)}(\xi)}{(t-\xi)^{q-n+1}}d\xi, \quad\quad n-1 \lt q \lt n . \end{eqnarray}$

(2.1)

Here, we give the function cascade synchronization method to fractional-order chaotic systems. Take a fractional-order dynamical system:

$\begin{eqnarray} \frac{d^q x}{d t^q} = f(x) = L x+N( x) \end{eqnarray}$

(2.2)

as a drive system. In the above $x = (x_1, x_2, x_3)^T$ is the state vector, $f: R^3 \rightarrow R^3$ is a continuous function, $L x$ and $N(x)$ represent the linear and nonlinear part of $f(x)$ , respectively.

Firstly, on copying any two equations of $(2.2)$ , such as the first two, one will obtain a sub-response system:

$\begin{eqnarray} \frac{d^q y}{d t^q} = L_1 y+N_1(y, x_3)+\tilde{U} \end{eqnarray}$

(2.3)

with $y = (X_1, Z)^T$ . In the above, $x_3$ is a signal provided by (2.2), while $\tilde{U} = (u_1, u_2)^T$ is a controller to be devised.

For the purpose of realizing the synchronization, we now define the error vector function via

$\begin{eqnarray} \tilde{e} = y-\tilde{Q}(\tilde{x})\tilde{x} \end{eqnarray}$

(2.4)

where $\tilde{e} = (e_1, e_2)^T$ , $\tilde{x} = (x_1, x_2)^T$ and $\tilde{Q}(\tilde{x}) = {\rm diag}({Q}_1(x_1), {Q}_2(x_2))$ .

Definition 1. For the drive system (2.2) and response system (2.3), one can say that the synchronization is achieved with a scaling function matrix $\tilde{Q}(\tilde{x})$ if there exists a suitable controller $\tilde{U}$ such that

$\begin{eqnarray} \lim\limits_{t \to \infty }{||\tilde{e}||} = \lim\limits_{t \to \infty }{||y-\tilde{Q}(\tilde{x})\tilde{x}||} = 0. \end{eqnarray}$

(2.5)

Remark 1. We would like to point out that one can have various different choices on the scaling function $\tilde{Q}(\tilde{x})$ , such as constant, power function, trigonometric function, hyperbola function, logarithmic and exponential function, as well as limited quantities of combinations and composite of the above functions. Particularly, when $\tilde{Q}(\tilde{x}) = I$ and $-I$ ( $I$ being a unit matrix), the problem is reducible to the complete synchronization and anti-phase synchronization of fractional-order chaotic systems, respectively. When $\tilde{Q}(\tilde{x}) = \alpha I$ , it becomes to the project synchronization. And when $\tilde{Q}(\tilde{x})$ = ${\rm diag}(\alpha_1, \alpha_2)$ , it turns to the modified projective synchronization. Hence, our method is more general than the existing methods in ^[4,13].

It is noticed from (2.5) that the system (2.3) will synchronize with (2.2) if and only if the error dynamical system (2.5) is stable at zero. For this purpose, an appropriate controller $\tilde{U}$ such that (2.5) is asymptotical convergent to zero is designed, which is described in the following theorem.

Theorem 1. For a scaling function matrix $\tilde{Q}(\tilde{x})$ , the FCS will happen between (2.2) and (2.3) if the conditions:

(i) the controller $\tilde{U}$ is devised by

$\begin{eqnarray} \tilde{U} = \tilde{K}\tilde{e}-N_1(y, x_3)+\tilde{Q}(\tilde{x})N_1(\tilde{x})+\tilde{P}(\tilde{x})\tilde{x} \end{eqnarray}$

(2.6)

(ii) the matrix $\tilde{K}$ is a $2\times 2$ matrix such that

$\begin{eqnarray} L_1+\tilde{K} = -\tilde{C}, \end{eqnarray}$

(2.7)

are satisfied simultaneously. In the above, $\tilde{P}(\tilde{x}) = {\rm {diag}}(\dot{Q}_1(x_1)\frac{ d^q x_1}{d t^q}, \dot{Q}_2(x_2)\frac{ d^q x_2}{d t^q})$ , $\tilde{K}$ is a $2\times 2$ function matrix to be designed. While $\tilde{C} = (\tilde{C}_{ij})$ is a $2\times 2$ function matrix wherein

$\begin{eqnarray} \tilde{C}_{ii} \gt 0 \quad \quad{\rm and} \quad \quad \tilde{C}_{ij} = -\tilde{C}_{ji}, \quad i\neq j. \end{eqnarray}$

(2.8)

Remark 2. It needs to point out that the construction of the suitable controller $\tilde{U}$ plays an important role in realizing the synchronization between (2.2) and (2.3). Theorem 2 provides an effective way to design the controller. It is seen from the theorem that the controller $\tilde{U}$ is closely related to the matrix $\tilde{C}$ . Once the condition $(2.8$ ) is satisfied, one will has many choices on the controller $\tilde{U}$ .

Remark 3. Based on the fact that the fractional orders themselves are varying parameters and can be applied as secret keys when the synchronization algorithm is adopted in secure communications, it is believed that our method will be more suitable for some engineering applications, such as chaos-based encryption and secure communication.

Proof: Let's turn back to the error function given in (2.4). Differentiating this equation with respect to $t$ and on use of the first two equations of (2.2) and (2.3), one will obtain the following dynamical system

$\begin{eqnarray} \frac{d^q \tilde{e}}{d t^q}& = &\frac{d^q y}{d t^q}-\tilde{Q}(\tilde{x})\frac{d^q x}{d t^q}-\tilde{P}(\tilde{x})\tilde{x}\\ & = &L_1 y+N_1(y, x_3)+\tilde{U}-\tilde{Q}(\tilde{x})[L_1 \tilde{x}+N_1(\tilde{x})]-\tilde{P}(\tilde{x})\tilde{x}\\ & = &L_1\tilde{e}+N_1(y, x_3)-\tilde{Q}(\tilde{x})N_1(\tilde{x})-\tilde{P}(\tilde{x})\tilde{x}+\tilde{K}\tilde{e}-N_1(y, x_3)+\tilde{Q}(\tilde{x})N_1(\tilde{x})+\tilde{P}(\tilde{x})\tilde{x}\\ & = &(L_1+\tilde{K})\tilde{e}. \end{eqnarray}$

(2.9)

Assuming that $\lambda$ is an arbitrary eigenvalue of matrix $L_1+\tilde{K}$ and its eigenvector is recorded as $\eta$ , i.e.

$\begin{eqnarray} ( L_1+\tilde{K} )\eta = \lambda \eta, \quad \eta\neq 0. \end{eqnarray}$

(2.10)

On multiplying (2.10) by $\eta^{\mathbf {H}}$ on the left, we obtain that

$\begin{eqnarray} \eta^{\mathbf {H}}( L_1+\tilde{K} )\eta = \lambda \eta^{\mathbf {H}}\eta \end{eqnarray}$

(2.11)

where ${\mathbf {H}}$ denotes conjugate transpose. Since $\bar{\lambda}$ is also an eigenvalue of $L_1+\tilde{K}$ , we have that

$\begin{eqnarray} \eta^{\mathbf {H}}( L_1+\tilde{K} )^{\mathbf {H}} = \bar{\lambda} \eta^{\mathbf {H}}. \end{eqnarray}$

(2.12)

On multiplying (2.12) by $\eta$ on the right, we derive that

$\begin{eqnarray} \eta^{\mathbf {H}}( L_1+\tilde{K} )^{\mathbf {H}}\eta = \bar{\lambda} \eta^{\mathbf {H}}\eta \end{eqnarray}$

(2.13)

From (2.11) and (2.13), one can easily get that

$\begin{eqnarray} \lambda+\bar{\lambda}& = &\eta^{\mathbf {H}}[( L_1+\tilde{K} )^{\mathbf {H}}+( L_1+\tilde{K} )]\eta/\eta^{\mathbf {H}}\eta\\[0.06in] & = &-\eta^{\mathbf {H}}(\tilde{C}+\tilde{C}^{\mathbf {H}})\eta/\eta^{\mathbf {H}}\eta\\[0.06in] & = &-\eta^{\mathbf {H}}\Lambda\eta/\eta^{\mathbf {H}}\eta \end{eqnarray}$

(2.14)

with $\Lambda = \tilde{C}+\tilde{C}^{\mathbf {H}}$ . Since $\tilde{C}$ satisfy the condition (2.8), one can know that $\Lambda$ denotes a real positive diagonal matrix. Thus we have $\eta^{\mathbf {H}}\Lambda\eta > 0$ . Accordingly, we can get

$\begin{eqnarray} \lambda+\bar{\lambda} = 2\mathbf {Re}(\lambda) = -\eta^{\mathbf {H}}\Lambda\eta/\eta^{\mathbf {H}}\eta \lt 0, \end{eqnarray}$

(2.15)

which shows

$\begin{eqnarray} |\mathbf{arg}\lambda| \gt \frac{\pi}{2} \gt \frac{q\pi}{2}. \end{eqnarray}$

(2.16)

According to the stability theorem in Ref. ^[34], the error dynamical system (2.9) is asymptotically stable, i.e.

$\begin{eqnarray} \lim\limits_{t \to \infty }{||\tilde{e}||} = \lim\limits_{t \to \infty }{||y-\tilde{Q}(\tilde{x})\tilde{x}||} = 0, \end{eqnarray}$

(2.17)

which implies that synchronization can be achieved between (2.2) and (2.3). The proof is completed.

Next, on copying the last two equations of (2.2), one will get another sub-response system:

$\begin{eqnarray} \frac{d^q z}{d t^q} = L_2 z+N_2(z, X_1)+\bar{U} \end{eqnarray}$

(2.18)

where $X_1$ is a synchronized variable in (2.3), $z = (X_2, X_3)^T$ and $\bar{U} = (u_3, u_4)^T$ is the controller being designed.

Here, we make analysis analogous to the above. Now we define the error $\bar{e}$ via

$\begin{eqnarray} \tilde{e} = z-\bar{Q}(\bar{x})\bar{x} \end{eqnarray}$

(2.19)

where $\bar{e} = (e_3, e_4)^T$ , $\bar{x} = (x_2, x_3)^T$ and $\bar{Q}(\bar{x}) = {\rm diag}({Q}_3(x_2), {Q}_4(x_3))$ . If devising the the controller $\bar{U}$ as

$\begin{eqnarray} \bar{U} = \bar{K}\bar{e}-N_2(z, X_1)+\bar{Q}(\bar{x})N_2(\bar{x})+\bar{P}(\bar{x})\bar{x} \end{eqnarray}$

(2.20)

and $L_2+\bar{K}$ satisfying

$\begin{eqnarray} L_2+\bar{K} = -\bar{C} \end{eqnarray}$

(2.21)

where $\bar{P}(\bar{x}) = {\rm {diag}}(\dot{Q}_3(x_2)\frac{ d^q x_2}{d t^q}, \dot{Q}_4(x_3)\frac{ d^q x_3}{d t^q})$ , $\bar{C} = (\bar{C}_{ij})$ denotes a $2\times 2$ function matrix satisfying

$\begin{eqnarray} \bar{C}_{ii} \gt 0 \quad \quad{\rm and} \quad \quad \bar{C}_{ij} = -\bar{C}_{ji}, \quad i\neq j, \end{eqnarray}$

(2.22)

then the error dynamical system (2.19) satisfies

$\begin{eqnarray} \lim\limits_{t \to \infty }{||\bar{e}||} = \lim\limits_{t \to \infty }{||z-\bar{Q}(\bar{x})\bar{x}||} = 0. \end{eqnarray}$

(2.23)

Therefore, one achieve the synchronization between the system (2.2) and (2.18). Accordingly, from (2.5) and (2.23), one can obtain that

$\begin{eqnarray} \left\{\begin{array}{l} \lim\limits_{t \to \infty }{||X_1-Q_1(x_1)x_1||} = 0, \\[0.08in] \lim\limits_{t \to \infty }{||X_2-Q_3(x_2)x_2||} = 0, \\[0.08in] \lim\limits_{t \to \infty }{||X_3-Q_4(x_3)x_3||} = 0. \end{array}\right. \end{eqnarray}$

(2.24)

which indicates the FCS is achieved for the fractional order chaotic systems.

3. Applications of FCS method to fractional-order unified chaotic system

In the sequel, we shall extend the applications of FCS approach to the fractional-order unified chaotic system to test the effectiveness.

The fractional-order unified chaotic system is described by:

$\begin{eqnarray} \left\{\begin{array}{l} \\ \frac{d^{q} x_{1}}{d t^q} = (25a+10)(x_2-x_1), \\[0.1in] \\ \frac{d^q x_{2}}{d t^q} = (28-35a)x_1-x_1x_3+(29a-1)x_2, \\[0.1in] \\ \frac{d^q x_{3}}{d t^q } = x_1x_2-\frac{a+8}{3}x_3, \end{array}\right. \end{eqnarray}$

(3.1)

where $x_i, (i = 1, 2, 3)$ are the state parameters and $a\in[0, 1]$ is the control parameter. It is know that when $0\leq a < 0.8$ , the system (3.1) corresponds to the fractional-order Lorenz system ^[35]; when $a = 0.8$ , it is the Lü system ^[36]; while when $0.8 < a < 1$ , it turns to the Chen system ^[37].

According to the FCS method in section 2, we take (3.1) as the drive system. On copying the first two equation, we get a sub-response system of (3.1):

$\begin{eqnarray} \left\{\begin{array}{l} \\ \frac{d^q X_{1}}{d t^q} = (25a+10)(Z-X_1)+u_1, \\[0.12in] \\ \frac{d^q Z}{d t^q} = (28-35a)X_1-Z x_3+(29a-1)Z+u_2, \end{array}\right. \end{eqnarray}$

(3.2)

where $\tilde{U} = (u_1, u_2)^T$ is a controller to be determined. In the following, we need to devise the desired controller $\tilde{U}$ such that (3.1) can be synchronized with (3.2). For this purpose, we set the error function $\tilde{e} = (e_1, e_2)$ via :

$\begin{eqnarray} \tilde{e} = (e_1, e_2) = (X_1-x_1(x_1^2+\alpha_1), Z-x_2\tanh{x_2}). \end{eqnarray}$

(3.3)

On devising the controller $\tilde{U}$ as (2.6), one can get that the error dynamical system is

$\begin{eqnarray} \frac{d^q \tilde{e}}{d t^q} = (L_1+\tilde{K})\tilde{e}, \end{eqnarray}$

(3.4)

where

$\begin{equation} \begin{split} & L_1 = \left( \begin{array}{cc} -10-25a & -10-25a\\ 28-35a & 29a-1 \\ \end{array} \right), \end{split} \quad\quad\quad \begin{split} & N_1(y, x_3) = \left( \begin{array}{c} 0 \\ -X_1x_3 \\ \end{array} \right){.} \end{split} \end{equation}$

(3.5)

If choosing, for example, the matrix $\tilde{K}$ as

$\begin{equation} \begin{split} & \tilde{K} = \left( \begin{array}{cc} -\lambda_1+25a+10 & x_1+x_1x_2-25a\\ -x_1-x_1x_2+35a-38 & -\lambda_2-29a+1 \\ \end{array} \right), \end{split} \end{equation}$

(3.6)

where $\lambda_1 > 0$ and $\lambda_2 > 0$ , then one can obtain that

$\begin{equation} \begin{split} & \tilde{C} = \left( \begin{array}{cc} -\lambda_1 & x_1+x_1x_2+10\\ -x_1-x_1x_2-10 & -\lambda_2 \\ \end{array} \right). \end{split} \end{equation}$

(3.7)

Therefore the dynamical system (3.4) becomes

$\begin{eqnarray} \frac{d^q \tilde{e}}{d t^q} = \begin{split} &\left( \begin{array}{cc} -\lambda_1 & x_1+x_1x_2\\ -x_1-x_1x_2 & -\lambda_2 \\ \end{array} \right)\tilde{e}. \end{split} \end{eqnarray}$

(3.8)

According to Theorem 2, the synchronization is realized in the system (3.1) and (3.2).

Subsequently, on copying the last two equations of (3.1), we get another sub-response system:

$\begin{eqnarray} \left\{\begin{array}{l} \\ \frac{\partial^q X_{2}}{\partial t^q} = (28-35a)X_1-X_1 X_3+(29a-1)X_2+u_3, \\[0.15in] \\ \frac{\partial^q X_{3}}{\partial t^q} = X_1X_2-\frac{a+8}{3}X_3+u_4, \end{array}\right. \end{eqnarray}$

(3.9)

where $\bar{U} = ({u_3, \ u_4})^T$ is the controller needed. When choosing the error function $\bar{e} = (e_3, e_4)$ as:

$\begin{eqnarray} \bar{e} = (e_3, e_4) = (X_2-\alpha_2 x_2, X_3-x_3(\alpha_3+e^{-x_3})), \end{eqnarray}$

(3.10)

and the controller $\bar{U}$ as (2.20), where

$\begin{equation} \begin{split} & L_2 = \left( \begin{array}{cc} 29a-1 & 0\\ 0 & -\frac{a+8}{3} \\ \end{array} \right), \end{split} \quad\quad\quad \begin{split} & N_2(z, X_1) = \left( \begin{array}{c} -X_1X_3 \\ X_1X_2 \\ \end{array} \right), \end{split} \end{equation}$

(3.11)

and the matrix $\bar{K}$ is chosen by

$\begin{equation} \begin{split} & \bar{K} = \left( \begin{array}{cc} -\lambda_3-29a+1 & 1+x_2x_3+e^{-x_3}\\ -1-x_2x_3-e^{-x_3} & -\lambda_4-\frac{a+8}{3} \\ \end{array} \right), \end{split} \end{equation}$

(3.12)

where $\lambda_3 > 0$ and $\lambda_4 > 0$ . Calculations show that the error dynamical system (2.19) becomes

$\begin{eqnarray} \frac{d^q \bar{e}}{d t^q} = \begin{split} &\left( \begin{array}{cc} -\lambda_3 & 1+x_2x_3+e^{-x_3}\\ -1-x_2x_3-e^{-x_3} & -\lambda_4 \\ \end{array} \right)\bar{e}. \end{split} \end{eqnarray}$

(3.13)

which, according to the stability theorem, indicates that $\bar{e}$ will approach to zero with time evolutions. Therefore, the FCS is realized for the fractional-order unified chaotic system.

In the above, we have revealed that the FCS is achieved for the fractional-order unified chaotic system in theory. In the sequel, we shall show that the FCS is also effective in the numerical algorithm.

For illustration, we set the fractional order $q = 0.98$ and the parameters $\lambda_i (i = 1, \cdots, 4)$ as $(\lambda_1, \lambda_2, \lambda_3, \lambda_4) = (2, 3, 0.5, 0.3)$ . It is noticed that when the value of $a\in[0, 1]$ is given, the system (3.1) will be reduced to a concrete system. For example, when $a = 0.2$ , it corresponds to the fractional-order Lorenz system. The chaotic attractors are depicted in Figure 1. Time responses of states variables and synchronization errors of the Lorenz system are showed in and , respectively. When $a = 0.8$ , it is the fractional-order Lü system. The chaotic attractors, time responses of state variables and synchronization errors are exhibited in –, respectively. When $a = 0.95$ , it turns to the fractional-order Chen system. Numerical simulation results are depicted in Figures 7–9. From the chaotic attractors pictures marked by Figures 1, 4 and 5, one can easily see that the trajectories of the response system (colored red) display certain consistency to that of the drive system (colored black) because of the special scaling functions chosen. Meanwhile, one can also see the synchronization is realized from Figures 3, 6 and 9. Therefore, we conclude that the FCS is a very effective algorithm for achieving the synchronization of the fractional-order unified chaotic system.

Figure 1. FCS of the fractional-order Lorenz system. Here we choose

$(\alpha_1, \alpha_2, \alpha_3) = (0.2, 2, -1.5)$ , initial values

$(x_1, x_2, x_3) = (-1, -0.5, -0.2)$ and

$(X_1, X_2, X_3) = (0.2, 0.3, 0.1)$ .

DownLoad: Full-Size Img PowerPoint

Figure 2. Time responses of state variables

$x_i$ and

$X_i (i = 1, 2, 3)$ for the fractional-order Lorenz system.

DownLoad: Full-Size Img PowerPoint

Figure 3. Synchronization errors of the Lorenz system.

DownLoad: Full-Size Img PowerPoint

Figure 4. FCS of the fractional-order Lü system with

$a = 0.8$ . Here we choose

$(\alpha_1, \alpha_2, \alpha_3) = (-0.5, 2.5)$ , initial values

$(x_1, x_2, x_3) = (0.5, -0.5, 0.2)$ and

$(X_1, X_2, X_3) = (0.15, -0.1, -0.1)$ .

DownLoad: Full-Size Img PowerPoint

Figure 5. Time responses of state variables

$x_i$ and

$X_i (i = 1, 2, 3)$ for the Lü system.

DownLoad: Full-Size Img PowerPoint

Figure 6. Synchronization errors of the Lü system.

DownLoad: Full-Size Img PowerPoint

Figure 7. FCS of the fractional-order Chen system with

$a = 0.95$ . Here we choose

$(\alpha_1, \alpha_2, \alpha_3) = (0.5, 1.5, 3)$ , initial values

$(x_1, x_2, x_3) = (1.5, 0.02, -0.01)$ and

$(X_1, X_2, X_3) = (-2, 0.01, -0.05)$ .

DownLoad: Full-Size Img PowerPoint

Figure 8. Time evolutions of state variables

$x_i$ and

$X_i$

$(i = 1, 2, 3)$ for the Chen system.

DownLoad: Full-Size Img PowerPoint

Figure 9. Synchronization errors of the Chen system.

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4. Conclusions

Chaos synchronization, because of the potential applications in telecommunications, control theory, secure communication et al, has attracted great attentions from various research fields. In the present work, via the stability theorem, we successfully extend the cascade synchronization of integer-order chaotic systems to a kind of general function cascade synchronization algorithm for fractional-order chaotic systems. Meanwhile, we apply the method to the fractional-order unified chaotic system for an illustrative test. Corresponding numerical simulations fully reveal that our method is not only accuracy, but also effective.

It is worthy of pointing out that the scaling function introduced makes the method more general than the complete synchronization, anti-phase synchronization, modified projective synchronization et al. Therefore, in this sense, our method is applicable and representative. However, the present work just study the fractional-order chaotic system without time-delay. It is known that in many cases the time delay is inevitably in the real engineering applications. Lag synchronization seems to be more practical and reasonable. Hence, it will be of importance and interest to study whether the FCS method can be used to realize the synchronization of fractional-order chaotic systems with time-delay. We shall considered it in our future work.

Acknowledgments

The authors would like to express their sincere thanks to the referees for their kind comments and valuable suggestions. This work is supported by the National Natural Science Foundation of China under grant No.11775116 and No.11301269.

Conflict of interest

We declare that we have no conflict of interests.

References

[1]	M. Tzitzeica, Sur une nouvelle classe des surfaces, CR Acad. Sci. Paris, 150 (1910), 955-956.
[2]	W. Rui, Exact traveling wave solutions for a nonlinear evolution equation of generalized tzitzéica-dodd-bullough-mikhailov type, J. Appl. Math., 2013 (2013), 1-14.
[3]	A. M. Wazwaz, The tanh method: Solitons and periodic solutions for the dodd-bullough-mikhailov and the tzitzeica-dodd-bullough equations, Chaos Soliton Fract., 25 (2005), 55-63. doi: 10.1016/j.chaos.2004.09.122
[4]	R. K. Dodd, R. K. Bollough, Polynomial conserved densities for the sine-gordon equations, Proc. R. Soc. Lond. A, 352 (1977), 481-503. doi: 10.1098/rspa.1977.0012
[5]	R. Conte, M. Musette, A. M. Grunland, Backlund transformation of partial differential equations from the painleve-gambier classification ii. tzitzeica equation, J. Math. Phys., 40 (1999), 2092-2106. doi: 10.1063/1.532853
[6]	J. Y. Zhu, X. G. Geng, Darboux transformation for tzitzeica equation, Commun. Theor. Phys., 45 (2006), 577-580. doi: 10.1088/0253-6102/45/4/001
[7]	A. V. Mikhailov, The reduction problem and the inverse scattering method, Physica, 1 (1981), 73-117.
[8]	O. H. El-Kalaawy, Exact soliton solutions for some nonlinear partial differential equations, Chaos Soliton Fract., 14 (2002), 547-552. doi: 10.1016/S0960-0779(01)00217-X
[9]	K. Khan, M. A. Akbar, Exact and solitary wave solutions for the tzitzeica-dodd-bullough and the modified kdv-zakharov-kuznetsov equations using the modified simple equation method, Ain Shams Eng. J., 4 (2013), 903-909. doi: 10.1016/j.asej.2013.01.010
[10]	R. Abazari, The (g'/g)-expansion method for tzitzéica type nonlinear evolution equations, Math. Comput. Model., 52 (2010), 1834-1845. doi: 10.1016/j.mcm.2010.07.013
[11]	J. Manafian, M. Lakestani, Dispersive dark optical soliton with tzitzéica type nonlinear evolution equations arising in nonlinear optics, Opt. Quant. Electron., 48 (2016), 116.
[12]	A. Korkmaz, K. Hosseini, Exact solutions of a nonlinear conformable time fractional parabolic equation with exponential nonlinearity using reliable methods, Opt. Quant. Electron., 49 (2017), 1-10. doi: 10.1007/s11082-016-0848-8
[13]	K. Hosseini, Z. Ayati, R. Ansari, New exact traveling wave solutions of the tzitzéica type equations using a novel exponential rational function method, Optik, 148 (2017), 85-89. doi: 10.1016/j.ijleo.2017.08.030
[14]	K. Hosseini, Z. Ayati, R. Ansari, New exact solution of the tzitzéica type equations in nonlinear optics using the exp_a function method, J. Mod. Opt., 65 (2018), 847-851. doi: 10.1080/09500340.2017.1407002
[15]	D. Kumar, K. Hosseini, F. Samadani, The sine-gordon expansion method to look for the traveling wave solutions of the tzitzéica type equations in nonlinear optics, Optik, 149 (2017), 439-446. doi: 10.1016/j.ijleo.2017.09.066
[16]	S. M. M. Alizamini, H. Rezazadeh, M. Eslami, et al. New extended direct algebraic method for the Tzitzica type evolution equations arising in nonlinear optics, Comp. Meth. Diff. Eq., 8 (2020): 28-53.
[17]	A. H. Arnous, A. R. Seadawy, R. T. Alqahtani, et al. Optical solitons with complex ginzburg-landau equation by modified simple equation method, Optik, 144 (2017), 475-480. doi: 10.1016/j.ijleo.2017.07.013
[18]	H. Rezazadeh, A. Korkmaz, M. Eslami, et al. Traveling wave solution of conformable fractional generalized reaction duffing model by generalized projective riccati equation method, Opt. Quant. Electron., 50 (2018), 1-13. doi: 10.1007/s11082-017-1266-2
[19]	K. Hosseini, P. Mayeli, R. Ansari, Bright and singular soliton solutions of the conformable time-fractional klein-gordon equations with different nonlinearities, Wave Random Complex., 28 (2018), 426-434.
[20]	A. Zafar, M. Raheel, A. Bekir, Exploring the dark and singular soliton solutions of biswas-arshed model with full nonlinear form, Optik, 204 (2020), 164133.
[21]	A. Korkmaz, Complex wave solutions to mathematical biology models i: Newell-whitehead-segel and zeldovich equations, J. Comput. Nonlin. Dyn., 13 (2018), 1-7.
[22]	A. Korkmaz, Exact solutions to (3+1) conformable time fractional jimbo-miwa, zakharov-kuznetsov and modified zakharov-kuznetsov equations, Commun. Theor. Phys., 67 (2017), 479.
[23]	Z. Ayati, K. Hosseini, M. Mirzazadeh, Application of kudryashov and functional variable methods to the strain wave equation in microstructured solids, Nonlinear Engineering, 6 (2017), 25-29.
[24]	K. Hosseini, J. Manafian, F. Samadani, et al. Resonant optical solitons with perturbation terms and fractional temporal evolution using improved tan(φ(η)/2)-expansion method and exp function approach, Optik, 158 (2018), 933-939. doi: 10.1016/j.ijleo.2017.12.139
[25]	K. Hosseini, P. Mayeli, D. Kumar, New exact solutions of the coupled sine- gordon equations in nonlinear optics using the modified kudryashov method, J. Mod. Optic., 65 (2018), 361-364. doi: 10.1080/09500340.2017.1380857
[26]	K. Hosseini, F. Samadani, D. Kumar, et al. New optical solitons of cubic-quartic nonlinear schr'odinger equation, Optik, 157 (2018), 1101-1105. doi: 10.1016/j.ijleo.2017.11.124
[27]	J. Vahidi, A. Zafar, A. Bekir, et al. The functional variable method to find new exact solutions of the nonlinear evolution equations with dual-power-law nonlinearity, Int. J. Nonlin. Sci. Num., (2020), article no. 10.1515/ijnsns-2019-0064.
[28]	M. S. Osman, H. Rezazadeh, M. Eslami, Traveling wave solutions for (3+1) dimensional conformable fractional zakharov-kuznetsov equation with power law nonlinearity, Nonlinear Engineering, 8 (2019), 559-567. doi: 10.1515/nleng-2018-0163
[29]	M. S. Osman., New analytical study of water waves described by coupled fractional variant boussinesq equation in fluid dynamics, Pramana, 93 (2019), 26.
[30]	V. S. Kumar, H. Rezazadeh, M. Eslami, et al. Jacobi Elliptic Function Expansion Method for Solving KdV Equation with Conformable Derivative and Dual-Power Law Nonlinearity, Int. J. Appl. Comput. Math., 5 (2019), 127.
[31]	A. Zafar, The expa function method and the conformable time-fractional kdv equations, Nonlinear Engineering, 8 (2019), 728-732. doi: 10.1515/nleng-2018-0094
[32]	H. Rezazadeh, A. Bekir, A. Zafar, et al. Exact solutions of (3 + 1)-dimensional fractional mkdv equations in conformable form via exp(-φ(τ))-expansion method, SN App. Sciences, 11 (2019), article no. 1436.
[33]	K. Hosseini, A. Zabihi, F. Samadani, et al. New explicit exact solutions of the unstable nonlinear schrödinger's equation using the exp_a and hyperbolic function methods, Opt. Quant. Electron., 50 (2018), article no. 82.
[34]	A. Zafar, A. Seadawy, The conformable space-time fractional mkdv equations and their exact solutions, J. King Saud Uni. Sci., 31 (2019), 1478-1484. doi: 10.1016/j.jksus.2019.09.003
[35]	A. H. Soliman, Abd-Allah Hyder, Closed-form solutions of stochastic KdV equation with generalized conformable derivatives, Phys. Scripta, 95 (2020), 065219.
[36]	Abd-Allah Hyder, A. H. Soliman, Exact solutions of space-time local fractal nonlinear evolution equations: A generalized conformable derivative approach, Results Phy., 17 (2020), 103135.
[37]	Abd-Allah Hyder, M. A. Barakat, General improved Kudryashov method for exact solutions of nonlinear evolution equations in mathematical physics, Phys. Scripta, 95 (2020), 045212.
[38]	H. A. Ghany, Abd-Allah Hyder, M. Zakarya, Exact solutions of stochastic fractional Korteweg de-Vries equation with conformable derivatives, Chinese Phys. B, 29 (2020), 030203.
[39]	A. Hyder, White noise theory and general improved Kudryashov method for stochastic nonlinear evolution equations with conformable derivatives, Adv. Differ. Equ., 236 (2020).
[40]	D. Lu, A. Seadawy, M. Arshad, Elliptic function solutions and travelling wave solutions of nonlinear dodd-bullough-mikhailov, two-dimensional sine-gordon and coupled schrödinger-kdv dynamical models, Results Phys., 10 (2018), 995-1005. doi: 10.1016/j.rinp.2018.08.001
[41]	A. Seadawy, S. Z. Alamri, Mathematical methods via the nonlinear two-dimensional water waves of olver dynamical equation and its exact solitary wave solutions, Results Phys., 8 (2018), 286-291. doi: 10.1016/j.rinp.2017.12.008
[42]	D. Lu, A. Seadawy, M. M. A. Khater, Bifurcations of new multi soliton solutions of the van der waals normal form for fluidized granular matter via six different methods, Results Phys., 7 (2017), 2028-2035. doi: 10.1016/j.rinp.2017.06.014
[43]	M. Iqbal, A. Seadawy, O. Khalil, et al. Propagation of long internal waves in density stratified ocean for the (2+1)-dimensional nonlinear nizhnik-novikov-vesselov dynamical equation, Results Phys., 16 (2020), article no. 102838.
[44]	A. Javid, N. Raza, M. S. Osman, Multi-solitons of thermophoretic motion equation depicting the wrinkle propagation in substrate-supported graphene sheets, Commun. Theor. Phys., 71 (2019), 362.
[45]	M. S. Osman, A. M. Wazwaz, A general bilinear form to generate different wave structures of solitons for a (3+1)-dimensional boiti-leon-manna-pempinelli equation, Math. Method Appl. Sci., 42, 2019.
[46]	M. S. Osman, D. Lu, M. M. A. Khater., A study of optical wave propagation in the nonautonomous schrödinger-hirota equation with power-law nonlinearity, Results Phys., 13 (2019), 102157.
[47]	D. Lu, M. S. Osman, M. M. A. Khater, et al. Analytical and numerical simulations for the kinetics of phase separation in iron (fe-cr-x (x=mo,cu)) based on ternary alloys, Phys. A, 537 (2020), 122634.
[48]	K. K. Ali, C. Cattani, J. F. Gómez-Aguilar, et al. Analytical and numerical study of the DNA dynamics arising in oscillator-chain of Peyrard-Bishop model, Chaos Soliton Fract., 139 (2020), 110089.
[49]	Y. Ding, M. S. Osman, A. M. Wazwaz, Abundant complex wave solutions for the nonautonomous Fokas-Lenells equation in presence of perturbation terms, Optik, 181 (2019), 503-513. doi: 10.1016/j.ijleo.2018.12.064
[50]	D. Lu, K. U. Tariq, M. S. Osman, et al. New analytical wave structures for the (3 + 1)-dimensional Kadomtsev-Petviashvili and the generalized Boussinesq models and their applications, Results Phys., 14 (2019), 102491.
[51]	I. Hamdy, A. Gawad, M. Osman, On shallow water waves in a medium with time-dependent dispersion and nonlinearity coefficients, J. Adv. Res., 6 (2015), 593-599. doi: 10.1016/j.jare.2014.02.004
[52]	M. S. Osman, Analytical study of solitons to benjamin-bona-mahony-peregrine equation with power law nonlinearity by using three methods, U.P.B. Sci. Bull., Series A, 80 (2018).
[53]	R. Nuruddeen, Multiple soliton solutions for the (3+1) conformable space-time fractional modified korteweg-de-vries equations, J. Ocean Eng. Sci., 3 (2018), 11-18. doi: 10.1016/j.joes.2017.11.004

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On Tzitéeica type nonlinear equations for multiple soliton solutions in nonlinear optics

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On Tzitéeica type nonlinear equations for multiple soliton solutions in nonlinear optics

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