Ultimate Boundedness and Finite Time Stability for a High Dimensional Fractional-Order Lorenz Model

Abstract

In this paper, the global attractive set (GAS) and positive invariant set (PIS) of the five-dimensional Lorenz model with the fractional order derivative are studied. Using the Mittag-Leffler function and Lyapunov function method, the ultimate boundedness of the proposed system are estimated. An effective control strategy is also designed to achieve the finite time stability of this fractional chaotic system. The corresponding boundedness and control scheme are numerically verified to show the effectiveness of the theoretical analysis.

1. Introduction

In recent years, interest in studying fractional-order dynamical systems has increased. Modeling many systems with fractional order equations is a necessity to study the behavior of dynamical systems in more realistic applications [1,2,3,4,5,6,7,8]. Integer order calculus is a special case in fractional calculus that is approximate to the real system in the mathematical model. Viscoelastic systems [9], distributed-order dynamical systems [10], hydro-turbine governing systems [11], and glucose-insulin regulatory system [12] are described by fractional order equations. Investigating the stability, control, and synchronization of chaotic fractional dynamical systems are three important issues that have been considered by researchers. Some physical systems exhibit fractional dynamic behavior due to their special properties. Since fractional calculus provides an accurate way to describe, predict, and control physical systems, it is used for these goals. The adaptive fuzzy control scheme, sliding-mode control, linear, nonlinear, active, feedback, and adaptive control method have been applied for the global stability and projective synchronization of chaotic fractional systems [13,14,15,16,17].

Chaos in dynamical systems presented with two-dimensional equations, including forced, dissipative Rayleigh-Benard convection. This system was named the 3-dimensional Lorenz model (3DLM) [18,19,20,21]. Lorenz showed that accurate weather predictions are impossible and numerical calculations indicated that the behavior of 3DLM becomes chaotic due to certain parameters and initial conditions.

Shen [22,23] studied the three-dimensional Lorenz model (3DLM) and developed it into a five-dimensional Lorenz model (5DLM). Many interesting properties of this system, which include improving the stability of the solutions [24], were investigated by Shen. He fully articulated the role of modes in the behavior of solutions and their stability. Additionally, the effect of some parameters in increasing the stability of system solutions was another highlight of his research. The Lorenz equations of 3, 5, and higher dimensions are derived from the Rayleigh-Benard convection equations, which have three physical processes, namely heating, dissipative, and nonlinear advection. Faghihi-Naini and Shen [25] examined the non-dissipative version of the 5DLM from another perspective. They proposed an analytical method for solving the 5DLM and thus were able to produce the quasi-periodic solutions to the system using the nonlinear feedback loop (NFL). These studies showed that by further expanding the NFL and introducing Lorenz systems with higher dimensions, interesting and practical results can be achieved.

Estimation of boundaries and attractive sets in dynamical systems is one of the important and practical issues in chaos control, chaos synchronization, Hausdorff dimension, and finding the system’s hidden attractors [26,27,28,29,30,31,32,33,34]. In fact, if one can calculate ultimate bound set (UBS) or globally attractive set (GAS) for a system, then one can claim that this system will not have any chaotic attractors, equilibrium points, periodic solutions, quasi-periodic solutions, etc., outside of these sets. This issue has a great application in controlling systems and preventing their possible problems [35,36,37,38]. In this paper, we will introduce the five-dimensional Lorenz model with the fractional order derivative. To the best of our knowledge, the GAS and the UBS for the fractional five-dimensional Lorenz model have not been investigated. As an innovation, in addition to proving the global boundness of the proposed system, we calculate a family of GASs. In fact, by changing system parameters and other conditions, we can create a variety of attractive sets. Furthermore, a controller is designed so that the controlled system can be stabilized in a finite time.

This article is organized in the following sections. Section 2 presents the integer and fractional model of five-dimensional Lorenz model (5DLM). In Section 3, we will study the Mittag-Leffler GASs of the fractional five-dimensional Lorenz model. Section 4 discusses the finite-time stability of the proposed system. Conclusions are drawn in Section 5.

2. Mathematical Model

In this section, we first review the process of generalizing the 3D Lorenz model to 5DLM and then introduce the fractional order five-dimensional Lorenz model.

2.1. Integer Order Five-Dimensional Lorenz Model

The flow occurred in a layer of fluid with uniform H depth was studied by Rayleigh [21]. Lorenz, in 1963 [18], introduced Rayleigh-Bennard equations for two-dimensional, forced, and dissipative convection:

$$\begin{array}{c}\hfill \frac{\partial }{\partial t}{\nabla }^2\psi =-\frac{\partial (\psi ,{\nabla }^2\psi )}{\partial (x,z)}+\nu {\nabla }^4\psi +g\gamma \frac{\partial \theta }{\partial x},\end{array}$$

(1)

$$\begin{array}{c}\hfill \frac{\partial }{\partial t}\theta =-\frac{\partial (\psi ,\theta )}{\partial (x,z)}+\frac{\Delta T}{H}\frac{\partial \psi }{\partial x}+\kappa {\nabla }^2\theta ,\end{array}$$

(2)

where $\psi$ is a stream function, $\theta$ is temperature perturbation, and the constants $g,\alpha ,\nu ,$ and $\kappa$ denote the acceleration of gravity, the coefficient of thermal expansion, the kinematic viscosity, and the thermal diffusivity, respectively. Additionally, $\Delta T$ is the difference in temperature between the top and bottom boundaries.

The chaotic system that Lorenz introduced is expressed as the following equations:

$$\begin{array}{ccc}\hfill & & {\dot{x}}_1=\sigma (x_2-x_1),\hfill \\ & & {\dot{x}}_2=-x_1{x}_3+r{x}_1-x_2,\hfill \\ & & {\dot{x}}_3=x_1{x}_2-b{x}_3.\hfill \end{array}$$

(3)

Although the 3DLM demonstrates solution dependence on initial conditions for chaotic solutions, the generalized Lorenz models (LMs) were derived for understanding the impact of mode truncations on solution stability and the route to chaos. Shen et al. [22] extended the 3DLM to the five-dimensional LM (5DLM) by including two additional Fourier modes with two additional vertical wave numbers. They used the five Fourier modes and rewrote $\psi$ and $\theta$ as the following:

$$\begin{array}{ccc}\hfill & & \psi =\kappa \frac{1+a^2}{a}\left(x_1{M}_1\right),\hfill \end{array}$$

(4)

$$\begin{array}{ccc}& & \theta =\frac{\Delta T}{\pi }\frac{R_c}{R_a}(x_2{M}_2-x_3{M}_3+x_4{M}_5-x_5{M}_6),\hfill \end{array}$$

(5)

where $R_a$ is Rayleigh number, $R_c$ is the critical value of free-slip Rayleigh-Benard problem, a is a ratio of the vertical scale of the convection cell to its horizontal scale, and

$$\begin{array}{ccc}& & M_1=\sqrt{2}sin\left(lx\right)sin\left(mz\right),\hfill \\ & & M_2=\sqrt{2}cos\left(lx\right)sin\left(mz\right),\hfill \\ & & M_3=sin\left(2mz\right),\hfill \\ & & M_5=\sqrt{2}cos\left(lx\right)sin\left(3mz\right),\hfill \\ & & M_6=sin\left(4mz\right),\hfill \end{array}$$

An additional mode $M_4=\sqrt{2}sin\left(lx\right)sin\left(3mz\right)$ is included to derive the 6DLM. Here, l and m are defined as $\frac{\pi a}{H}$ and $\frac{\pi }{H},$ representing the horizontal and vertical wave numbers, respectively, and a is a ratio of the vertical scale of the convection cell to its horizontal scale. The term H is the domain height and $\frac{2H}{a}$ represents the domain width.

By coordinate transformation, the original equation can be reduced to the following five-dimensional nonlinear dynamics:

$$\begin{array}{ccc}\hfill & & {\dot{x}}_1=\sigma (x_2-x_1),\hfill \\ & & {\dot{x}}_2=-x_1{x}_3+r{x}_1-x_2,\hfill \\ & & {\dot{x}}_3=x_1{x}_2-x_1{x}_4-b{x}_3,\hfill \\ & & {\dot{x}}_4=x_1{x}_3-2x_1{x}_5-d{x}_4,\hfill \\ & & {\dot{x}}_5=2x_1{x}_4-4b{x}_5,\hfill \end{array}$$

(6)

where $\sigma ,r,b,d$ are constant parameters. The dynamic properties and conditions of the new five-dimensional Lorenz model (6) were investigated in [22,25,39,40,41]. They discussed in detail the numerical solutions, which included chaotic, periodic, and quasi-periodic responses.

2.2. Basic Definitions of Fractional Calculus

Definition 1.

The Riemann-Liouville fractional integral function $X\left(t\right)$ is

$$\begin{array}{c}\hfill I^{\alpha }X\left(t\right)=\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^t{(t-s)}^{\alpha -1}X\left(s\right)ds,t>0,\end{array}$$

(7)

$\Gamma (.)$ is the gamma function:

$$\begin{array}{c}\hfill \Gamma \left(m\right)={\int }_0^{\infty }{s}^{m-1}{e}^{-s}ds.\end{array}$$

(8)

Definition 2.

The Caputo fractional-order derivative of function $X\in C^n([t_0,+\infty ),\mathbb{R})$ is

$$D_t^{\alpha }X\left(t\right)=\frac{1}{\Gamma (n-\alpha )}{\int }_0^t{(t-s)}^{n-\alpha -1}{X}^{\left(n\right)}\left(s\right)ds,$$

(9)

Definition 3.

The Mittag-Leffler function $E_{\alpha ,\beta }(.)$ with two parameters is defined as

$$\begin{array}{c}\hfill E_{p,q}\left(m\right)=\sum _{k=0}^{\infty }\frac{m^{\left(k\right)}}{\Gamma (kp+q)},\end{array}$$

(10)

where $p>0,q>0,$ and m is a complex number. It is clear that

$$\begin{array}{c}\hfill E_p\left(s\right)=E_{p,1}\left(m\right),E_{0,1}\left(m\right)=\frac{1}{1-m},E_{1,1}\left(m\right)=e^m.\end{array}$$

Let us consider the following fractional-order system

$$\begin{array}{c}\hfill D_t^{\alpha }X=f\left(X\right),X\left(t_0\right)=X_0,\end{array}$$

(11)

where $X\in {\mathbb{R}}^n$, $f:{\mathbb{R}}^n\to {\mathbb{R}}^n$ is sufficiently smooth and $X(t,t_0,X_0)$ is the solution.

Definition 4.

For a given Lyapunov function ${\mathcal{V}}_{\lambda }\left(t\right)={\mathcal{V}}_{\lambda }\left(X\left(t\right)\right)$ with $\lambda >0$, if there exists constants ${\mathcal{L}}_{\lambda }>0$ and $r_{\lambda }>0 for all X_0\in {\mathbb{R}}^n$ such that

$$\begin{array}{c}\hfill {\mathcal{V}}_{\lambda }\left(t\right)-{\mathcal{L}}_{\lambda }\le ({\mathcal{V}}_{\lambda }\left(t_0\right)-{\mathcal{L}}_{\lambda })E_{\alpha }\left(-r_{\lambda }{(t-t_0)}^{\alpha }\right),t\ge t_0,\end{array}$$

(12)

for ${\mathcal{V}}_{\lambda }\left(X\right)>{\mathcal{L}}_{\lambda }$, then ${\mathcal{U}}_{\lambda }=\{X|{\mathcal{V}}_{\lambda }\left(X\left(t\right)\right)\le {\mathcal{L}}_{\lambda }\}$ is said to be the Mittag-Leffler GAS of system (11). If for any $X_0\in {\mathcal{U}}_{\lambda }$ and any $t>t_0,X(t,t_0,X_0)\in {\mathcal{U}}_{\lambda }$, then ${\mathcal{U}}_{\lambda }$ is said to be a Mittag-Leffler PISs, where $X=X\left(t\right),X_0=X\left(t_0\right).$

2.3. Fractional Five-Dimensional Lorenz Model

Let us introduce the fractional calculus into the system (6). The fractional-order of the five-dimensional Lorenz model can be written as follows:

$$\begin{array}{ccc}\hfill & & D_t^{\alpha }{x}_1\left(t\right)=\sigma (x_2-x_1),\hfill \\ & & D_t^{\alpha }{x}_2\left(t\right)=-x_1{x}_3+r{x}_1-x_2,\hfill \\ & & D_t^{\alpha }{x}_3\left(t\right)=x_1{x}_2-x_1{x}_4-b{x}_3,\hfill \\ & & D_t^{\alpha }{x}_4\left(t\right)=x_1{x}_3-2x_1{x}_5-d{x}_4,\hfill \\ & & D_t^{\alpha }{x}_5\left(t\right)=2x_1{x}_4-4b{x}_5,\hfill \end{array}$$

(13)

when $\sigma =10,b=\frac{8}{3},r=25,d=\frac{19}{3}$, and $\alpha =0.9$, the chaotic behavior of system (6), are shown in Figure 1 and Figure 2. It is noteworthy that changing the order of fractional derivative changes the behavior of the system from chaotic to steady-state, which is shown in Figure 3. The time responses and phase portraits of system (13), with $\sigma =10,b=\frac{8}{3},r=85,d=\frac{19}{3}$, and $\alpha =0.98$, are shown in Figure 4 and Figure 5.

3. Mittag-Leffler GAS Estimation of the Fractional Five-Dimensional Lorenz Model

In this section, we will calculate the Mittag-Leffler GASs for the fractional-order of five-dimensional Lorenz model (13).

Lemma 1

([42]). If $X\left(t\right)\in \mathbb{R}$ is a continuous and differentiable function, then

$$\begin{array}{c}\hfill D^{\alpha }\left(X^2\left(t\right)\right)\le 2X\left(t\right)D^{\alpha }\left(X\right(t\left)\right),\end{array}$$

(14)

Lemma 2

([42]). For $\alpha \in (0,1)$ and constant $\overline{\omega }\in \mathbb{R}$, if a continuous function $g\left(t\right)$ meets

$$\begin{array}{c}\hfill D^{\alpha }\left(g\right(t\left)\right)\le \overline{\omega }g\left(t\right),t\ge 0,\end{array}$$

(15)

then

$$\begin{array}{c}\hfill g\left(t\right)\le g\left(0\right)E_{\alpha }\left(\overline{\omega }{t}^{\alpha }\right),t\ge 0.\end{array}$$

(16)

The following theorem investigated the Mittag-Leffler GASs and the Mittag-Leffler PISs of the system (13):

Theorem 1.

Let $\sigma >0,b>0,r>0,$ and $d>0$ denote

$$\begin{array}{c}\hfill {\mathcal{U}}_{\lambda ,\mu }=\left\{X\left(t\right)\in {\mathbb{R}}^5\mid \lambda x_1^2+\mu x_2^2+\mu {\left(x_3-\frac{\sigma \lambda +r\mu }{\mu }\right)}^2+\mu x_4^2+\mu {(x_5-\frac{\sigma \lambda +r\mu }{2\mu })}^2\le R_{max}^2\right\}.\end{array}$$

Then ${\mathcal{U}}_{\lambda ,\mu }$ is the Mittag-Leffler GASs and the Mittag-Leffler PISs of system (13), where

$$\begin{array}{c}\hfill R_{max}^2=\frac{2b{(\sigma \lambda +r\mu )}^2}{\mu \eta }\end{array}$$

(17)

and $\eta =min\{\sigma ,b,d,1\}>0.$

Proof. 

Define the following generalized positively definite and radically unbounded Lyapunov function

$$\begin{array}{ccc}\hfill & & {\mathcal{V}}_{\lambda ,\mu }(x_1,x_2,x_3,x_4,x_5)=\hfill \\ & & \frac{1}{2}\lambda x_1^2+\frac{1}{2}\mu x_2^2+\frac{1}{2}\mu {\left(x_3-\frac{\sigma \lambda +r\mu }{\mu }\right)}^2+\frac{1}{2}\mu x_4^2+\frac{1}{2}\mu {\left(x_5-\frac{\sigma \lambda +r\mu }{2\mu }\right)}^2,\hfill \end{array}$$

(18)

where $\lambda >0,$$\mu >0.$ Computing the derivative of ${\mathcal{V}}_{\lambda ,\mu }$ along the trajectory of system (13), we have

$$\begin{array}{ccc}\hfill & & D^{\alpha }{\mathcal{V}}_{\lambda ,\mu }\left(X\left(t\right)\right)\le \lambda x_1{D}^{\alpha }{x}_1+\mu x_2{D}^{\alpha }{x}_2+\mu \left(x_3-\frac{\sigma \lambda +r\mu }{\mu }\right)D^{\alpha }{x}_3\hfill \\ & & +\mu x_4{D}^{\alpha }{x}_4+\mu \left(x_5-\frac{\sigma \lambda +r\mu }{2\mu }\right)D^{\alpha }{x}_5\hfill \\ & & =\lambda x_1\left(-\sigma x_1+\sigma x_2\right)+\mu x_2(-x_1{x}_3+r{x}_1-x_2)\hfill \\ & & +\mu \left(x_3-\frac{\sigma \lambda +r\mu }{\mu }\right)(x_1{x}_2-x_1{x}_4-b{x}_3)\hfill \\ & & +\mu x_4(x_1{x}_3-2x_1{x}_5-d{x}_4)+\mu \left(x_5-\frac{\sigma \lambda +r\mu }{2\mu }\right)(2x_1{x}_4-4b{x}_5)\hfill \\ & & =-\frac{1}{2}\sigma \lambda x_1^2-\frac{1}{2}\mu x_2^2-\frac{1}{2}b\mu {\left(x_3-\frac{\sigma \lambda +r\mu }{\mu }\right)}^2-\frac{1}{2}d\mu x_4^2\hfill \\ & & -\frac{1}{2}4b\mu {\left(x_5-\frac{\sigma \lambda +r\mu }{2\mu }\right)}^2+F\left(X\right),\hfill \end{array}$$

where

$$\begin{array}{c}\hfill F\left(X\right)=-\frac{1}{2}\sigma \lambda x_1^2-\frac{1}{2}\mu x_2^2-\frac{1}{2}b\mu x_3^2-\frac{1}{2}d\mu x_4^2-\frac{1}{2}4b\mu x_5^2+\frac{b{(\sigma \lambda +r\mu )}^2}{\mu }.\end{array}$$

(19)

It is obvious that ${\displaystyle F\left(X\right)\le \underset{X\in {\mathbb{R}}^5}{sup}F\left(X\right)=l_{\lambda ,\mu }=\frac{b{(\sigma \lambda +r\mu )}^2}{\mu }.}$

From there we have

$$\begin{array}{c}\hfill D^{\alpha }{\mathcal{V}}_{\lambda ,\mu }\left(X\left(t\right)\right)\le -\eta V_{\lambda ,\mu }+l_{\lambda ,\mu },\end{array}$$

(20)

i.e.,

$$\begin{array}{c}\hfill D^{\alpha }({\mathcal{V}}_{\lambda ,\mu }\left(t\right)-\frac{l_{\lambda ,\mu }}{\eta })\le -\eta ({\mathcal{V}}_{\lambda ,\mu }\left(t\right)-\frac{l_{\lambda ,\mu }}{\eta }).\end{array}$$

(21)

Based on Lemma 2, one can obtain

$$\begin{array}{c}\hfill {\mathcal{V}}_{\lambda ,\mu }\left(t\right)-\frac{l_{\lambda ,\mu }}{\eta }\le ({\mathcal{V}}_{\lambda ,\mu }\left(0\right)-\frac{l_{\lambda ,\mu }}{\eta })E_{\alpha }(-\eta t^{\alpha }),t\ge 0.\end{array}$$

(22)

Based on Definition 4, from (22), one can conclude that the ellipsoid ${\mathcal{U}}_{\lambda ,\mu }$ for $\sigma >0$, $b>0$, $r>0,$ and $d>0$ is a Mittag-Leffler GAS and Mittag-Leffler PIS for the system (13). This completes the proof. □

Remark 1.

(i) If we take $\lambda =1,\mu =1,$ then

$$\begin{array}{c}\hfill {\mathcal{U}}_{1,1}=\left\{(x_1,x_2,x_3,x_4,x_5)|x_1^2+x_2^2+{(x_3-(\sigma +r))}^2+x_4^2+{(x_5-\frac{\sigma +r}{2})}^2\le 2b{(\sigma +r)}^2\right\},\end{array}$$

is the Mittag-Leffler GAS of system (13).

When $\sigma =10,b=\frac{8}{3},r=25,$ and $d=\frac{19}{3},$ we have

$$\begin{array}{c}\hfill {\mathcal{U}}_{1,1}=\left\{(x_1,x_2,x_3,x_4,x_5)|x_1^2+x_2^2+{(x_3-35)}^2+x_4^2+{(x_5-\frac{35}{2})}^2\le {\left(80.8\right)}^2\right\}.\end{array}$$

Figure 6 shows the phase portraits and the Mittag-Leffler GAS of system (13) in the different spacess defined by ${\mathcal{U}}_{1,1}.$

By considering the value of $r=85,$ the solutions of (13) change from steady-state to chaotic. Figure 7 shows chaotic attractors of system (13) in the different spacess defined by ${\mathcal{U}}_{1,1}.$(ii) Let us take $\lambda =1,\mu =2,$ then we get that the set

$$\begin{array}{ccc}& & {\mathcal{U}}_{1,2}=\hfill \\ & & \left\{(x_1,x_2,x_3,x_4,x_5)|x_1^2+2x_2^2+2{(x_3-\frac{\sigma +2r}{2})}^2+2x_4^2+2{(x_5-\frac{\sigma +2r}{4})}^2\le b{(\sigma +2r)}^2\right\},\hfill \end{array}$$

is the Mittag-Leffler GAS of system (6).

When $\sigma =10,b=\frac{8}{3},r=25,$ and $d=\frac{19}{3},$ we have

$$\begin{array}{c}\hfill {\mathcal{U}}_{1,2}=\left\{(x_1,x_2,x_3,x_4,x_5)|x_1^2+2x_2^2+2{(x_3-30)}^2+2x_4^2+2{(x_5-15)}^2\le 97.9^2\right\}.\end{array}$$

Figure 8 shows the phase portraits and the Mittag-Leffler GAS of system (13) in the different spacess defined by ${\mathcal{U}}_{1,2}.$

(iii) Let us take $\lambda =2,\mu =1,$ then we get that the set

$$\begin{array}{c}\hfill {\mathcal{U}}_{2,1}=\left\{(x_1,x_2,x_3,x_4,x_5)|2x_1^2+x_2^2+{(x_3-(2\sigma +r))}^2+x_4^2+{(x_5-\frac{2\sigma +r}{2})}^2\le 2b{(2\sigma +r)}^2\right\},\end{array}$$

is the Mittag-Leffler GAS of system (6).

When $\sigma =10,b=\frac{8}{3},r=25,$ and $d=\frac{19}{3},$ we have

$$\begin{array}{c}\hfill {\mathcal{U}}_{2,1}=\left\{(x_1,x_2,x_3,x_4,x_5)|2x_1^2+x_2^2+{(x_3-45)}^2+x_4^2+{(x_5-22.5)}^2\le 103.9^2\right\}.\end{array}$$

Figure 9 shows the phase portraits and the Mittag-Leffler GAS of system (13) in the different spacess defined by ${\mathcal{U}}_{2,1}.$

4. Finite-Time Stabilization of Fractional Order System

In this section, we design an effective control scheme to stabilize the fractional-order five-dimensional Lorenz model in a finite time. The controlled fractional order system is given as

$$\begin{array}{ccc}\hfill & & D_t^{\alpha }{x}_1\left(t\right)=\sigma (x_2-x_1)+u_1,\hfill \\ & & D_t^{\alpha }{x}_2\left(t\right)=-x_1{x}_3+r{x}_1-x_2+u_2,\hfill \\ & & D_t^{\alpha }{x}_3\left(t\right)=x_1{x}_2-x_1{x}_4-b{x}_3+u_3,\hfill \\ & & D_t^{\alpha }{x}_4\left(t\right)=x_1{x}_3-2x_1{x}_5-d{x}_4+u_4,\hfill \\ & & D_t^{\alpha }{x}_5\left(t\right)=2x_1{x}_4-4b{x}_5+u_5,\hfill \end{array}$$

(23)

where $u_1,u_2,u_3,u_4$, and $u_5$ are control parameters of the system (23). Now the control goal is to design a suitable robust controller to stabilize system (23) around zero in finite time.

Lemma 3

([43]). If $x_1,x_2,\cdots ,x_n$ are positive real numbers, and $0<m<1,$ then we have the following inequality:

$$\begin{array}{c}\hfill (|x_1|+|x_2|+\cdots +|x_n{|)}^m\le |x_1{|}^m+|x_2{|}^m+\cdots +{|x_n|}^m.\end{array}$$

(24)

Theorem 2

([43]). Assume that $\Omega \subset {\mathbb{R}}^n$ is a domain containing the origin and $\mathcal{V}(t,X):[t_0,\infty )\times \Omega \to \mathbb{R}$ is a continuously differentiable function and locally Lipschitz, so that

$$\begin{array}{ccc}\hfill & & {\gamma }_1{(\Vert X\Vert }^a)\le \mathcal{V}(t,X)\le {\gamma }_2{(\Vert X\Vert }^{ab}),\hfill \end{array}$$

(25)

$$\begin{array}{ccc}\hfill & & D^{\alpha }\mathcal{V}(t,X)\le -{\gamma }_3{(\Vert X\Vert }^{ab})\hfill \end{array}$$

(26)

where $0<\alpha <1,a>0,b>0,{\gamma }_i(i=1,2,3)>0,$ then the system (23) is called Mittag-Leffler stable.

In the previous section, we have derived the Mittag-Leffler GAS and Mittag-Leffler PIS for the system (13):

$$\begin{array}{c}\hfill |x_1|\le \frac{R_{max}}{\sqrt{\lambda }},|x_2|\le \frac{R_{max}}{\sqrt{\mu }},|x_3-\frac{\sigma \lambda +r\mu }{\mu }|\le \frac{R_{max}}{\sqrt{\mu }},|x_4|\le \frac{R_{max}}{\sqrt{\mu }},|x_5-\frac{\sigma \lambda +r\mu }{2\mu }|\le \frac{R_{max}}{\sqrt{\mu }},\end{array}$$

where $R_{max}$ was given in (17). This implies that there exists constants $K_1$ and $K_2$ such that $K_1\le \Vert X\Vert \le K_2.$

Theorem 3.

The controlled chaotic system (23) can be finite-timely stabilized by the controller,

$$\begin{array}{ccc}\hfill & & u_1=\sigma x_1-r{x}_2-\frac{k}{2}{x}_1^s,\hfill \\ & & u_2=x_2-\sigma x_1-\frac{k}{2}{x}_2^s,\hfill \\ & & u_3=b{x}_3-d{x}_4-\frac{k}{2}{x}_3^s,\hfill \\ & & u_4=d{x}_4+d{x}_3-b{x}_5-\frac{k}{2}{x}_4^s,\hfill \\ & & u_5=4b{x}_5+b{x}_4-\frac{k}{2}{x}_5^s,\hfill \end{array}$$

(27)

where $0<s<1,k>0$, and the finite time T is estimated by:

$$\begin{array}{c}\hfill T\le {\left(x_1^2+\cdots +x_5^2\right)}^{\alpha -\frac{1+s}{2}}(0,X)\frac{\Gamma (1-\frac{1+s}{2})\Gamma (1+\alpha )}{k\Gamma (\alpha -\frac{1+s}{2}+1)}.\end{array}$$

(28)

Proof. 

To prove the stability of system (23), let us use the classic Lyapunov direct method, proposing the quadratic function as a Lyapunov candidate, which is positive definite

$$\begin{array}{c}\hfill \mathcal{V}(T,X)=x_1^2+x_2^2+x_3^2+x_4^2+x_5^2.\end{array}$$

(29)

Substituting (27) into (23), calculating the $\alpha$-order fractional derivative of the Lyapunov function and using Lemma 1, one can get

$$\begin{array}{ccc}\hfill & & D^{\alpha }\mathcal{V}(T,X)\le 2x_1{D}^{\alpha }{x}_1+2x_2{D}^{\alpha }{x}_2+2x_3{D}^{\alpha }{x}_3+2x_4{D}^{\alpha }{x}_4+2x_5{D}^{\alpha }{x}_5\hfill \\ & & =2x_1\left(-\sigma x_1+\sigma x_2+u_1\right)+2x_2(-x_1{x}_3+r{x}_1-x_2+u_2)\hfill \\ & & +2x_3(x_1{x}_2-x_1{x}_4-b{x}_3+u_3)\hfill \\ & & +2x_4(x_1{x}_3-2x_1{x}_5-d{x}_4+u_4)+2x_5(2x_1{x}_4-4b{x}_5+u_5)\hfill \\ & & =-k{x}_1^{1+s}-k{x}_2^{1+s}-k{x}_3^{1+s}-k{x}_4^{1+s}-k{x}_5^{1+s},\hfill \end{array}$$

then,

$$\begin{array}{c}\hfill D^{\alpha }(x_1^2+\cdots +x_5^2)\le -k{(x_1^2+x_2^2+\cdots +x_5^2)}^{\frac{1+s}{2}}.\end{array}$$

(30)

It is obvious that the Lyapunov function defined in (29) satisfies conditions (25) and (26) in Theorem 2. Thus, the system (23) is Mittag-Leffler stable. Based on the property of Caputo fractional derivatives,

$$\begin{array}{c}\hfill D^{\alpha }{X}^n=\frac{\Gamma (n+1)}{\Gamma (n+1-\alpha )}{X}^{n-\alpha }{D}^{\alpha }X,\end{array}$$

(31)

we have

$$\begin{array}{c}\hfill D^{\alpha }{\left(x_1^2+\cdots +x_5^2\right)}^{\alpha -\frac{1+s}{2}}=\frac{\Gamma (\alpha -\frac{1+s}{2}+1)}{\Gamma (1-\frac{1+s}{2})}{\left(x_1^2+\cdots +x_5^2\right)}^{-\frac{1+s}{2}}{D}^{\alpha }(x_1^2+\cdots +x_5^2),\end{array}$$

(32)

therefore, we obtain from (30) and (32)

$$\begin{array}{c}\hfill D^{\alpha }{\left(x_1^2+\cdots +x_5^2\right)}^{\alpha -\frac{1+s}{2}}\le -k\frac{\Gamma (\alpha -\frac{1+s}{2}+1)}{\Gamma (1-\frac{1+s}{2})},\end{array}$$

(33)

and one may take integral of both sides of (33) from 0 to T as follows:

$$\begin{array}{c}\hfill {\left(x_1^2+\cdots +x_5^2\right)}^{\alpha -\frac{1+s}{2}}(T,X)-{\left(x_1^2+\cdots +x_5^2\right)}^{\alpha -\frac{1+s}{2}}(0,X)\le -k\frac{\Gamma (\alpha -\frac{1+s}{2}+1)}{\Gamma (1-\frac{1+s}{2})\Gamma (1+\alpha )}{T}^{\alpha }.\end{array}$$

(34)

The time is expressed as

$$\begin{array}{c}\hfill T\le {\left[{\left(x_1^2+\cdots +x_5^2\right)}^{\alpha -\frac{1+s}{2}}(0,X)\frac{\Gamma (1-\frac{1+s}{2})\Gamma (1+\alpha )}{k\Gamma (\alpha -\frac{1+s}{2}+1)}\right]}^{\frac{1}{\alpha }}.\end{array}$$

(35)

□

Numerical results are presented to show the effectiveness of the designed controller in the fractional order 5DLM. We set the time step size to 0.001, and the order of fractional derivative is selected as $\alpha =0.98,$ and $\alpha =0.7.$ The states of the system under the controller are depicted in Figure 10 and Figure 11, which indicate that the trajectories of the system can be stabilized to the origin in a finite time. In addition, the chaotic behavior is suppressed. Numerical results show that changing $\alpha$ and k causes the system converges to zero quickly.

5. Conclusions

In this paper, we introduced the fractional-order five-dimensional Lorenz model. Using the Lyapunov function and fractional-order derivative, the Mittag-Leffler GASs and Mittag-Leffler PISs for this system are obtained. Furthermore, we investigated finite-time stabilization of the fractional-order five-dimensional Lorenz model. Finally, simulation results were given to show the validity of the proposed scheme. The estimation of the bound sets of fractional-order chaotic systems was not seriously investigated due to their specific complexities. Therefore, the calculation of global attractive set (GAS) and positive invariant set (PIS) for fractional 5DLM can encourage us to provide new methods for estimating the bounds of various fractional order systems.