On Generalized Fractional Dynamical System With Order Lying in (0, 2): Stability Analysis, Chaotic Behaviour, Control and Synchronization

The generalized fractional dynamical system with order lying in (0, 2) is investigated. We present the stability analysis of that system using Mittag-Leﬄer function, the Gronwall-Bellman Lemma and Laplace transform. The bifurcation diagram of generalized fractional-order Chen system is given. We investigate a theorem to control the chaotic generalized fractional-order systems by linear feedback control. Two examples to achieve the theorem of control are given. The synchronization between two diﬀerent chaotic generalized fractional systems is presented. We give a theorem to calculate the control functions which achieve synchronization. This theorem is applied to achieve the synchronization between diﬀerent generalized fractional-order systems with order lying in (0, 1]. And, also, used to achieve the synchronization between the identical generalized fractional-order L¨u systems with order lying in [1, 2). There exist an agreement among analytical results and numerical treatments for stability, control and synchronization theorems.

The generalized fractional derivative has many features over the integer derivatives and has potential many applications. Since chaos in fractional-order models is more complicated than the integer cases, then its are C D α,ρ x(t) = Ax(t) + f (x(t)), (1.1) where 0 < α < 2, ρ ≥ 0 and C D α,ρ denotes the left generalized Caputo-type fractional derivative. On the other hand, for the numerical simulation purposes of fractional order models, the predictor-corrector (P-C) techniques are one of the most efficient, stable and accurate methods that was implemented and modified to numerically solve Caputo fractional differential equations [31]. Odibat and Baleanu [32] present an adaptive predictor corrector method for the numerical solution of generalized Caputo-type initial value problems. In order to construct the predictor-corrector method for the IVP (1.1), we will follow the same procedure as in [32].
The pervious papers on the stability of generalized fractional-order nonlinear systems are investigated with fractional-order lying in (0, 1). In this paper, we stated the generalized fractional dynamical system with order in (0, 2). The stability analysis of that system using Mittag-Leffler function, the Gronwall-Bellman Lemma and Laplace transform is illustrated. We show that chaotic solutions for generalized fractional-order models are more complicated than the classical fractional and integer cases. We controlled the chaotic generalized fractionalorder systems using the linear feedback control. By this technique of control, we synchronized two different chaotic generalized fractional-order systems. The paper is outlined as follows: In Section 2, we address some important preliminaries. In Section 3, using Mittag-Leffler function, the Gronwall-Bellman Lemma and Laplace transform, we prove the solution of the generalized fractional dynamical system approach to zero at the infinity.
We investigate the control of chaotic generalized fractional system by linear feedback control in Section 4. We give two examples to achieve the control theorem . In Section 5, the control functions which achieve synchronized two different chaotic generalized fractional systems is illustrated. The synchronization between the different generalized fractional-order Chen and Lü systems is presented. And the synchronization between the identical generalized fractional-order Lü systems is introduced. Finally, a conclusion is given to summary our works.

Liouville-Caputo Fractional Calculus
The left-sided and right-sided Riemann-Liouville integrals of order α, when 0 < α < 1, are defined, respec- and where Γ represents the Euler Gamma function. The corresponding inverse operators, i.e., the left-sided and right-sided fractional derivatives of order α, are then defined based on (2.1) and (2.2), as and This allows for the definition of the left and right Riemann-Liouville fractional derivatives of order α (n − 1 < α < n), n ∈ N as (2.5) and (2.6) Furthermore, the corresponding left-sided and right-sided Caputo derivatives of order α (n − 1 < α < n) are obtained as and (2.8) The Caputo operator satisfies the rule (2.10) The generalized left-sided Riemann-Liouville fractional derivatives of order α (n − 1 < α < n), n ∈ N is defined We can observe when ρ = 1, we recover the Riemann-Liouville fractional derivative in (2.5). Furthermore, the corresponding generalized left-sided Caputo derivatives of order α (n − 1 < α < n) are obtained as
Definition 2.2. (Jarad et al. [30]). The generalized left-sided Caputo derivative of f order α (n − 1 < α < n), n ∈ N is defined by (2.13) further analysis includes (2.14) We can observe when ρ = 1 in (2.13) , we recover the Caputo fractional derivative in (2.7). For the rest of this paper, the Laplace transform will be used to help us in studying stability of the generalized fractional differential equations. The ρ−Laplace transform was recently introduced in the literature [33]. The ρ−Laplace transform of the Caputo generalized fractional derivative is expressed in the following form: where C D α,ρ = C 0 D α,ρ x (i.e. left generalized Caputo-type fractional derivative of order α).
Besides, the ρ−Laplace transform of a given function f is described in the form: The two-parameter Mittag-Leffler function is defined as: α,β (−az α ) is: .
By using definition 2.3, the following relation is obtained as: From (2.20) one gets: where d m , m = 0, 1, ..., k are constants depend on β.

Stability analysis
Theorem 3.1. The zero solution of generalized Caputo fractional-order system (1.1) is stable if: where x ∈ R n×1 , A ∈ R n×n , t ∈ R + and λ i ( A ρ α ) be the eigenvalues of matrix A ρ α . Proof: Two cases will be considered separately.
(b) The case 1 < α < 2 In this case, the initial condition is We can get the solution of (1.1) with the initial condition (3.12) by using the ρ-Laplace transform and ρ-Laplace inverse transform as: By part 2 of Theorem 3.1, there exists C > 0 and δ 0 such that (3.14) Using Eqs. (3.3), (3.14) and Lemma 2.3, (3.13) gives x(s) ds.

Control of chaotic generalized fractional-order systems
We introduce a technique of control of solutions of chaotic generalized fractional-order systems by linear feedback control. The generalized fractional-order system (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. So, 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 take two examples of chaotic generalized fractional-order systems to achieve Theorem 4.1. The first example for order lying in (0, 1] and the other example for order lying in [1, 2).

Example 1
In this example, we do a control for chaotic generalized fractional-order Chen system [32] by linear feedback control. The chaotic generalized fractional-order Chen system can be written as: and solution x of that system is given in Fig. 2, which shows that system has chaotic attractor for the same parameters of Fig. 1. If we take ρ = 1 and the same values of other parameters and the initial conditions of Fig. 1, the behaviour of the solution of fractional-order Chen system can be shown in Fig. 3. We can notice that the solution of generalized fractional-order Chen system is more complicated than the solution of usual fractional-order Chen system for the same time (t=10). The chaotic Chen system (4.3) can be written after adding the control functions as: The control functions can be written as: (4.5)   Using (4.5), the control system (4.4) can be written as System (4.6) holds the sufficient conditions of Theorem 4.1 as: 1. The eigenvalues of A + K are λ 1 = −39.2, λ 2 = −23 and λ 3 = −2.8, then the zero solution of C D α,ρ x(t) = In numerical simulation, if we take the same values of the parameters and the initial values of Fig. 1, the solution of the controlled system (4.6) is approach to zero as shown in Fig. 4. This means there exist agreement between Theorem (4.1) and the numerical results.

Example 2
In this subsection, we use linear feedback control to control the solution of chaotic generalized fractional-order Lü system for order lying in [1,2). The chaotic generalized fractional-order Lü system takes the form: By adding control functions, system (4.7) can be written as We can write the control functions as (4.9) Using (4.9), the control system (4.8) can be written as where System (4.10) holds Theorem 4.1 as: 1. The eigenvalues of A + K are λ 1 = −46, λ 2 = −23 and λ 3 = −5, then the zero solution of C D α,ρ x(t) = The zero solution of controlled system (4.10) is asymptotically stable as shown in Fig. 6 for the same choice of parameters and initial values of Fig. 5.

Synchronization between different and identical chaotic generalized fractional-order system
In this section, we investigate the synchronization between two different chaotic generalized fractional-order systems using linear feedback control method. We present two examples of synchronization between chaotic generalized fractional-order systems. The first one discuss the synchronization between two different chaotic (c) Figure 6: The behaviour of chaotic system after adding the control (4.8) in (a) (t, generalized fractional-order Lü and Chen systems with order α ∈ (0, 1]. The second example explains the synchronization between two identical chaotic generalized fractional-order Lü systems with order α ∈ (1, 2]. Definition 5.1. We can said the drive system (1.1) is synchronized with the following response system C D α,ρ y = By + f (y) + u, From systems (1.1) and (5.1), the error system can be written as: Proof: Using the control functions (5.3), system (5.2) can be written as: by taking ρ-Laplace transform for system (5.4), then (5.5) then, then the synchronization between the drive system (1.1) and the response system (5.1) can be achieved. ✷

Synchronization between two different chaotic generalized fractional-order Lü and Chen systems
We consider the drive system is chaotic generalized fractional-order Lü system (4.7) and chaotic generalized fractional-order Chen system (4.4) is the response system. The response system after adding the control functions can be written as: C D α,ρ y 1 (t) = a 1 (y 2 − y 1 ) + u 1 , C D α,ρ y 2 (t) = (c 1 − a 1 )y 1 − y 1 y 3 + c 1 y 2 + u 2 , Using the drive system (4.7), the response system (5.8) and the control functions (5.3), the error system takes the form  Fig. 7 shows the same chaotic attractor for drive system (4.7) and response system (5.8), while the synchronization errors go to zero as given in Fig. 8.

Synchronization between two identical chaotic generalized fractional-order Lü systems
In this subsection, we present the identical synchronization between two generalized fractional Lü systems with order α ∈ (1, 2]. We consider the chaotic Lü system (4.7) the drive system and the response takes the form: C D α,ρ y 1 (t) = a(y 2 − y 1 ) + u 1 , C D α,ρ y 2 (t) = −y 1 y 3 + cy 2 + u 2 , C D α,ρ y 3 (t) = y 1 y 2 − by 3 + u 3 . (5.10) Using Theorem 5.1, the error system is   9 shows the state variables of drive system (4.7) and response system (5.10) versus t. The synchronization errors approach to zero as shown in Fig. 10. The generalized fractional dynamical system was simulated using Adams-Bashforth-Moulton method in this paper.

Conclusion
We introduced the generalized fractional dynamical system with order in (0,2). In Theorem 3.1, the stability analysis of that system is investigated using Mittag-Leffler function, the Gronwall-Bellman Lemma and Laplace transform. The chaotic generalized fractional-order Chen and Lü systems and the bifurcation diagram Compliance with ethical standards

Conict of Interest:
We have no conict of interest. Declarations: Not applicable. Figure 1 See the Manuscript Files section for the complete gure caption.        See the Manuscript Files section for the complete gure caption.

Figure 10
See the Manuscript Files section for the complete gure caption.