Dynamics , Chaos Control , and Synchronization in a Fractional-Order Samardzija-Greller Population System with Order Lying in ( 0 , 2 )

Department of Statistics and Operations Researches, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt Department of Basic Engineering Sciences, College of Engineering, Majmaah University, Al-Majmaah 11952, Saudi Arabia Mathematics Department, College of Sciences and Humanities Studies Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj, Saudi Arabia Mathematics Department, Faculty of Science, Hail University, Hail 2440, Saudi Arabia

Chaos synchronization and control in dynamical systems are essential applications of chaos theory.They have become focal topics for research since the elegant work of Ott et al. in chaos control [28] and the pioneering work of Pecora and Carroll in chaos synchronization [29].Chaos control is sometimes needed to refine the behavior of a chaotic model and to remove unexpected performance of power electronics.Synchronization of chaos has also useful applications to biological, chemical, physical systems and secure communications.Furthermore, synchronization and control in chaotic fractional-order dynamical systems have been investigated by authors [30][31][32].
The integer-order Samardzija-Greller population model is a system of ODEs that generalizes the Lotka-Volterra equations and expresses the behaviors of two species predatorprey population dynamical system.This model was proposed by Samardzija and Greller in 1988 [33].They had proved the existence of complex oscillatory behavior in this model.In 1999, chaos synchronization had firstly been investigated in this model by Costello [34].Oancea et al. utilized this system to achieve the pest control in agricultural systems [35].In 2018, Elsadany et al. proved the existence of generalized Hopf (Bautin) bifurcation in this model [36].Recently, some works investigating synchronization in the fractionalorder Samardzija-Greller model with order less than one have appeared [37][38].Up to the present day, dynamics, chaos control, and synchronization have not been investigated in the fractional-order Samardzija-Greller system with order lying in (0, 2).
So, our objective in this paper is to investigate the rich dynamics and achieve chaos synchronization and control in the fractional-order Samardzija-Greller model with order lying in (0, 2).Moreover, applying the fractional Routh-Hurwitz criteria [23,39] enables us to illustrate the role of the fractional parameter on synchronizing and controlling chaos in this model.Motivated by the previous discussion, numerical verifications are performed to show the existence of wide range of chaotic dynamics in this model using the aids of phase portraits and Lyapunov exponents.

Mathematical Preliminaries
Here, Caputo definition [17] is adopted as where α ∈ ℜ + , f l denotes the ordinary lth derivative of f t , l is an integer that satisfies l − 1 < α ≤ l, and the operator J β is defined by where Γ is defined as Therefore, fractional modeling provides more accuracy in both theoretical and experimental results of the ecological model which are naturally related with long range memory behavior which is very important in modeling ecological systems.Thus, increasing the range of the fractional parameter α from the interval (0, 1) that is commonly used in most literatures to the interval (0, 2) provides greater degrees of freedom in modeling the population system.In addition, increasing the interval of fractional parameter increases the complexity in the system since the fractional parameter α is in a close relationship with fractals which are abundant in ecosystems.Furthermore, as α is increased, more adequate description of the whole time domain of the process is achieved and the system may show rich variety of complex behaviors such as sensitive dependence on initial conditions.Consider the following fractional-order system: which represents a linearized form of the nonlinear system: Moreover, the following inequalities [40] arg determine the local stability of a steady state X ∈ ℜ n of the fractional-order system (4), where λ i is any eigenvalue of the matrix J.The stability region for α ∈ 0, 1 and α ∈ 1, 2 is shown in Figures 1(a) and 1(b), respectively.Moreover, stability conditions and their applications to nonlinear systems of FODEs were investigated in [23,41].

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where the prey of the population is denoted by x and the predators of the population are denoted by y and z.The predators y, z do not interact directly with one another but compete for prey x, and a, b, c are nonnegative parameters used to discuss the bifurcation phenomena in the model as stated by Samardzija and Greller in 1988 [33].Furthermore, the positive value of the parameter a indicates that two different predators y and z consume the prey x.However, if parameter a is vanished, a classical version of Lotka-Volterra model is obtained.By replacing first-order derivatives in system (7) with fractional derivatives of order 0 < α < 2, we obtain So, a greater degree of accuracy can be obtained in our population model by using the property of evolution (8) due to the existence of fractional derivatives which is essential to the proposed Samardzija-Greller model The discriminant of P λ is given by If all roots of (10) satisfy the inequalities (6), then the steady state of the linearized form of system ( 9) is locally asymptotically stable (LAS).To discuss the local stability of X * , we prove the following theorem: , where J is the Jacobian matrix computed at the steady state X * .Proof 1.Let s i be a very small perturbation defined as s i = x i − x * i , where i = 1,2,3.Therefore, (9), with derivatives in the Caputo sense, can be written as with initial values s i 0 = x i 0 − x * i .A linearization of the above equation, based on the Taylor expansion, can take the form where α ∈ 0, 2 , J is computed at the steady state X * = x * 1 , x * 2 , x * 3 , and s = s 1 s 2 s 3 T .Hence, where Q = diag λ 1 , λ 2 , λ 3 , λ i is an eigenvalue of J, and P is the corresponding eigenvector.Suppose that γ = P −1 s = γ 1 γ 2 γ 3 T , then it follows that The above system can easily be solved by the aid of Mittag-Leffler functions as follows If the eigenvalue λ i satisfies the condition arg λ i J X * > απ/2, then the perturbations s i t are decreasing which implies that X * is LAS.
(i) The steady state X * is LAS iff

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provided that the discriminant of ( 10) is positive.
(iii) If the discriminant of ( 10) is negative, and r 1 > 0, Hence, the following lemmas are provided.
Lemma 3 (see [42]).Assume that system ( 5) is written in the form where α ∈ 0, 2 , C ∈ ℜ n×n is a constant matrix, and h X t is a nonlinear vector function such that lim where represents the l 2 -norm, then the zero steady state of system ( 5) is LAS provided that arg

Stability Conditions for the
Steady State E 0 .The characteristic equation of the steady state E 0 is described as It is easy to check that r 3 = −b < 0, so by applying the stability condition (iv), we conclude that the steady state E 0 of system (8) is unstable.

Stability Conditions for the
Steady State E 1 .The characteristic equation of the steady state E 1 is described as So by utilizing the FRH conditions (i)-(iv), we obtain the following results: (1) If the discriminant of ( 21) is negative; the steady state (2) The steady state E 1 of system ( 8) is LAS only if b > a.
Remark 1.The stability conditions (i) and (iii) are not satisfied for the steady state E 1 .

Stability Conditions for the
Steady State E 2 .The characteristic equation of the steady state E 2 is described as where m = b/a.So by utilizing the FRH conditions (i)-(iv), we obtain the following results: (1) When the discriminant of ( 22) is positive, the steady state (2) When the discriminant of ( 22) is negative, the steady state (3) The steady state E 2 of system ( 8) is LAS only if b < a.

Hopf Bifurcation in Samardzija-Greller Model (8)
Although exact periodic solutions do not exist in autonomous fractional systems [43], some asymptotically periodic signals converge to limit cycles have been observed by numerical simulations in many fractional systems.These limit cycles attract the nearby solutions of such systems.Therefore, Hopf bifurcation (HB) in autonomous fractional systems is expected to occur around an equilibrium point of such systems if it has a pair of complex conjugate eigenvalues and at least one negative eigenvalue.However, determining the precise bifurcation type is difficult.In Samardzija-Greller system (8), the occurrence of HB is expected around E 1 = 1, 1 + c, 0 at the critical parameter Complexity at which the transversality condition holds and 2 1 − 2 < c < 2 1 + 2 , a < b.For α = 0 98, a = 0 8, b = 3, and c = 0 07, system (8) converges to a limit cycle as shown in Figure 2.

Chaos in Samardzija-Greller Model (8)
Based on the algorithm given in [44] and using the parameter set a, b, c = 8/10, 3, 3 , the maximum values of Lyapunov exponents (MLEs) of system (8) are computed.The calculations of the MLEs give the values 0.21315, 0.038468, 0.0089, and 0.1562 when α = 1 1, α = 1 05, α = 1 00, and α = 0 99, respectively.In Figure 3, it is shown that the chaotic attractor of system (8) related to the fractional-order case is more complicated than the attractor related to the integer-order case.
Another parameter set a, b, c = 3, 4, 10 is used to explore a variety of complex dynamics and new chaotic regions in the Samardzija-Greller model (7) and its corresponding fractional-order form.The results are depicted in Figure 4 which shows that the complex dynamics of the system exist in a wide range of the fractional parameter α.These foundations are confirmed by the calculation of maximal Lyapunov exponents which are depicted in Figure 5.

Chaos Control in System (8)
In the following, we will control chaos in system (8) when 1 < α < 2 using linear control technique and also we will discuss the role of fractional parameter α on controlling chaos in Samardzija-Greller model using a linear feedback control technique.

Chaos Control of Samardzija-Greller System (8) with α
Lying in (1, 2) via Linear Control.Assume that the controlled Samardzija-Greller system is described by Thus, system (25) is rewritten as where is a constant parameter matrix, and the linear controller U is written as = MKX, 28 5 Complexity where U is to be designed later, and h X t is also defined as So, according to Lemma 3, the zero steady state x − x * , y − y * , z − z * of the controlled system ( 26) is LAS if the linear control matrix is chosen such that arg The controlled Samardzija-Greller model ( 25) is numerically integrated as a = 3, b = 4,c = 10, and α = 1 1 and linear control functions where k 1 , k 2 , k 3 ≥ 0 and ε is a real constant.Thus, the selection k 1 > 1, k 2 > 0, k 3 > 0 ensures that the conditions arg λ i C − U > απ/2, i = 1,2,3, hold.Figures 6(a)-6(c) show that system (26) is controlled to the steady states E 0 , E 1 , E 2 as using (k 1 = 100, k 2 = 150, k 3 = 100, and ε = 100), (k 1 = 100, k 2 = 150, k 3 = 150, and ε = 0 3), (k 1 = 100, k 2 = 100, k 3 = 150, and ε = 0 3), respectively.(8) via the FRH Criterion.The FRH criterion is employed to control chaos in system (8) using linear feedback control technique which is more easy of implementation, cheap in cost, and 6 Complexity more appropriate of being designed in real-world applications.Moreover, we will illustrate the role of fractional parameter α on controlling chaos in system (8).Consider the following system:

Chaos Synchronization of Samardzija-Greller System (8) via Nonlinear Feedback Control
The drive system is introduced as 39 and the response system is presented as where α ∈ 0, 2 and v 1 , v 2 , v 3 are the nonlinear feedback controllers.Then, we suppose that By subtracting (39) from (40) and using (41), we get and the FCGs must satisfy Proof 2. The Jacobian matrix of system (42) with the controllers (43) evaluated at the origin steady state is described by So, if we select k 1 > 1, k 2 > 0, k 3 > 0, then all the eigenvalues of the Jacobian matrix (46) have negative signs.Thus, according to condition (6), the origin steady state of system (42) is LAS for 0 < α < 2. Consequently, the proof is completed.
The fractional-order systems (39) and ( 40) are numerically integrated using the system's parameters a = 8/10, b = 3, and c = 3, the controllers (43) with k 1 = 20, k 2 = 20, k 3 = 20, and the fractional parameters α = 0 99 and α = 1 1, respectively.The results are depicted in Figure 10.Proof 3. The Jacobian matrix of system (42) with the controllers (47) evaluated at the origin steady state is described by and has the following characteristic equation:   49) satisfies the fractional Routh-Hurwitz condition (iii), since its discriminant is negative, a 1 ∈ ℜ + , a 2 ∈ ℜ + , and a 3 = a 1 a 2 .So according to Lemma 2, the origin steady state of system (42) is LAS only for all α ∈ 0, 1 .Consequently, the states of Samardzija-Greller model (39) asymptotically approach the states of Samardzija-Greller model (40) as the controllers (47) are employed.Since the characteristic equation ( 49) has a pair of purely imaginary eigenvalues, the condition of asymptotic stability of the origin steady state 0,0,0 of ( 42) is not satisfied as 1 ≤ α < 2. Thus, the theorem is now proved.
Therefore, as the parameter α is decreased, the nonlinear feedback control technique has a stabilizing effect on the synchronization process in the chaotic Samardzija-Greller systems.These results are illustrated in Figure 11.

Concluding Remarks
In this paper, nonlinear dynamics, conditions of chaos control, and synchronization are studied in the fractional Samardzija-Greller population model with fractional order between zero and two.To the best of the authors' knowledge, dynamical behaviors and chaos applications in this model have not been investigated elsewhere when the fractional order lies between zero and two.This kind of study has great importance to predator-prey ecosystems since it provides greater degrees of freedom in modeling such systems.In addition, increasing the interval of fractional parameter helps to raise the complexity in the system since the fractional parameter is in a close relationship with fractals which are abundant in ecosystems.Furthermore, as the fractional parameter α is increased, more accurate description of the whole time domain of the process is achieved and the system will show rich dynamics such as the existence of chaos.12 Complexity By using the fractional Routh-Hurwitz criterion, local stability in this model has been demonstrated as α ∈ 0, 2 .Moreover, the chaoticity of this model has been numerically examined by calculating the maximal Lyapunov exponents using the Wolf's algorithm.The calculations show that chaos exists in this system when the fractional parameter lies in the intervals (0, 1) and (1,2).
Chaos control in this system has been achieved via a novel linear control technique as α ∈ 1, 2 .Furthermore, the role of fractional parameter α on controlling chaos in this model has also been illustrated using a linear feedback control technique.It has been proved that using the appropriate FCGs, the trajectory of Samardzija-Greller population model is stabilized to the steady state E 1 = 1, 1 + c, 0 as the fractional parameter α lies between zero and one.
Chaos synchronization has been achieved in this system via nonlinear feedback control method.It has also been shown that, under certain choice of nonlinear controllers, the fractional-order Samardzija-Greller model is synchronized only when the fractional parameter α is less than one.Thus, the obtained results illustrate that the parameter α has also a stabilizing role on the synchronization process in this system.
All the analytical results have been verified using numerical simulations.

Figure 2 :
Figure 2: The trajectory of system (8) with the parameter set a, b, c = 0 8,3,0 07 and fractional parameter α = 0 98 is attracted by a limit cycle: (a) view in xyz-space; (b) the solution versus time.