Mittag–Leffler stability, control, and synchronization for chaotic generalized fractional-order systems

In this paper, we investigate the generalized fractional system (GFS) with order lying in (1,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(1, 2)$\end{document}. We present stability analysis of GFS by two methods. First, the stability analysis of that system using the Gronwall–Bellman (G–B) Lemma, the Mittag–Leffler (M–L) function, and the Laplace transform is introduced. Secondly, by the Lyapunov direct method, we study the M–L stability of our system with order lying in (1,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(1, 2)$\end{document}. Using the modified predictor–corrector method, the solutions of GFSs are calculated and they are more complicated than the classical fractional one. Based on linear feedback control, we investigate a theorem to control the chaotic GFSs with order lying in (1,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(1, 2)$\end{document}. We present an example to verify the validity of control theorem. We state and prove a theorem to calculate the analytical formula of controllers that are used to achieve synchronization between two different chaotic GFSs. An example to study the synchronization for systems with orders lying in (1,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(1, 2)$\end{document} is given. We found an agreement between analytical results and numerical simulations.


Introduction
Fractional derivatives and fractional differential equations have been shown to be a beneficial approach in physical phenomena modeling in different fields of engineering and science [1]. It is worth mentioning that many practical systems display memory and genetic characteristics, which can be better described by fractional calculus than by integer calculus, such as electromagnetic waves [2], hydroturbine-governing systems [3], viscoelastic systems [4], financial systems [5], and wind-turbine generators [6]. Also, many different definitions of fractional derivatives are introduced according to different kernels [7][8][9]. Theses definitions are used in Riesz-space fractional equations [10], time-fractional differential equations [11][12][13], complex network [14,15], materials constitutive equations [16,17], fractional diffusion modeling [18][19][20], fractional love models [21,22], and control theories [23][24][25]. Furthermore, many types of synchronization such as complete, anti, where C D σ ,ρ denotes the left generalized Caputo-type fractional derivative, 1 < σ < 2, ρ ≥ 0, x ∈ R n is a state variable, A is a (n × n) constant matrix, and f (x(t)) ∈ R n is a vector of continuous functions. The predictor-corrector (P-C) method is a numerical simulation purposed for Caputo fractional-order systems [39]. Odibat and Baleanu [40] introduced a modified P-C method for the numerical solution of generalized Caputo-type IVPs. We use the same procedure as in [40] to solve system (1.1). Chaos synchronization and control theory in fractional calculus have become important topics in recent years. Several types of synchronization and different control techniques are used in many applications such as neural networks [41][42][43], biology [44,45], and secure communications [46,47]. The generalized fractional-order systems will be used in potential applications as an extension of classical fractional-and integer-order ones.
The previous papers on studying the stability of GFSs investigated with fractional-order lying in (0, 1), while this paper states the stability of GFSs with order lying in (1,2). The stability analysis of that system using the G-B Lemma, the M-L function, and the Laplace transform is illustrated. The M-L stability of GFS with order lying in (1, 2) is investigated by the Lyapunov direct method. We show that chaotic solutions for GF models are more complicated than the classical fractional and integer cases [31]. We controlled the chaotic GFSs using linear feedback control [48]. By this technique of control, we synchronized two different chaotic GFSs. The paper is outlined as follows: In Sect. 2, we address some important preliminaries. In Sect. 3, using the G-B Lemma, the M-L function, and the Laplace transform, we prove the solution of the generalized fractional dynamical system on its approach to zero at infinity. By linear feedback control, a theorem to control chaotic GFS is investigated in Sect. 4. We give an example to test the validity of this theorem.
In Sect. 5, the control functions that are used to achieve synchronization between two different chaotic GFSs are illustrated. An example of synchronization between different GFSs with orders lying in (1, 2) is presented. Finally, the conclusion of our work is given in Sect. 6.
The ρ-Laplace transform of a function g is given as: (2.13)

Stability analysis
In this section, we introduce two methods to study the stability of system (1.1) for σ ∈ (1, 2). The first one depends on the Gronwall-Bellman (G-B) Lemma, the Mittag-Leffler (M-L) function, and the Laplace transform, while the second method depends on the Lyapunov direct method and it is called M-L stability.

Theorem 3.1 The zero solution of a generalized Caputo fractional-order system
Proof The initial conditions of system (1.1) are given as: Using the ρ-Laplace transform and ρ-Laplace inverse transform, one obtains the solution of (1.1) with the initial conditions (3.1) as follows: By part 2 of Theorem 3.1, there exists C > 0 and δ 0 such that Using Eq. (3.3) and Lemma 2.3, (3.2) gives x(s) ds.

The Mittag-Leffler (M-L) stability for system (1.1)
The definition of M-L stability and its theorem stability for the case σ ∈ (0, 1) is introduced [34]. In this subsection, we present the definition of M-L stability for the case σ ∈ (1, 2).
A theorem to prove that the solution of system (1.1) is M-L stable is investigated using the Lyapunov direct method.

Control of chaotic generalized fractional systems (GFSs)
We introduce a technique to control the solutions of chaotic GFSs by linear feedback control. The GFS (1.1) can be written after adding the vector of control functions u(t) as: We can present the linear feedback control functions as u(t) = Kx(t), where K is n × n constant matrix. Hence, the controlled system (4.1) becomes: We investigate the sufficient conditions to hold that system (4.2) is asymptotically stable in the following theorem.
Proof The proof is similar to that of Theorem 3.1.
We give an example of chaotic GFSs with order lying in (1,2) to test the validity of Theorem 4.1.

An example
In this subsection, we control the solution of chaotic GF Lü system with order lying in (1, 2) using linear feedback control. The chaotic GF Lü system takes the form:  Figs. 1(a), (b), and (c) that correspond to the values ρ = 1, ρ = 2.5, and ρ = 3, respectively. Figure 1(a) shows the chaotic behavior of the classical fractional-order Lü system (ρ = 1), while the chaotic behaviors of generalized fractionalorder Lü systems for ρ = 2.5, and ρ = 3 are shown in Figs. 1(b) and (c). This means that the complicated solution behavior of system (4.3) depends on the value of parameter ρ.
By adding control functions, system (4.3) can be written as We can write the control functions as Using (4.5), the system (4.4) is obtained where System (4.6) holds Theorem 4.1 as: The zero solution of system (4.6) is asymptotically stable, as shown in Fig. 2 for the same choice of parameters and initial values as in Fig. 1(b).

Synchronization between two different chaotic generalized fractional systems (GFSs)
In this section, we introduce the synchronization between two different chaotic GFSs using a linear feedback control method. We present an example to verify the validity of the proposed theorem of synchronization.
Definition 5. 1 We can say that the drive system (1.1) is synchronized with the following response system C D σ ,ρ y = By + f y(t) + u, is a (n × n) constant matrix and u ∈ R n is a vector of control functions.
From systems (1.1) and (5.1), the error system can be written as: We investigate a theorem to calculate the analytical formula of control functions that achieve synchronization between two different chaotic GFSs.   then the synchronization between the drive system (1.1) and the response system (5.1) can be achieved. Page 13 of 16

Figure 5
The synchronization errors of the drive system (4.3) and the response system (5.9) in (a) (t, e 1 ) diagram, (b) (t, e 2 ) diagram, and (c) (t, e 3 ) diagram In the numerical treatment, the values of the parameters and the initial conditions of the drive system (4.3) and for the response system (5.9) are the same values that are taken in Fig. 1(b) and Fig. 3, respectively. The synchronization is achieved and the results are shown in Figs. 4 and 5. Figure 4 shows the same chaotic attractor for drive system (4.3) and response system (5.9), while the synchronization errors approach zero, as given in Fig. 5.

Conclusion
We introduced the generalized fractional dynamical system with order in (1,2). In Theorem 3.1, the stability analysis of that system is investigated using the Mittag-Leffler function, the Gronwall-Bellman Lemma, and the Laplace transform. We proposed M-L stability of our system with order lying in (1, 2) based on the Lyapunov direct method in Theorem 3.2. The chaotic GF Lü and Lorenz systems are presented. Using linear feedback control, we illustrated the control of chaotic GFS in general and an example is given to test the validity of Theorem 4.1. We investigated the synchronization between two different chaotic GFSs. The analytical formula of the control functions (5.3) that achieve synchronization are given in Theorem 5.1. Synchronization between the different GF Lü and Lorenz systems is achieved. Other examples of GFSs can be similarly studied.