Computation of Stability Criterion for Fractional Shimizu–Morioka System Using Optimal Routh–Hurwitz Conditions

: Nowadays, the dynamics of non-integer order system or fractional modelling has become a widely studied topic due to the belief that the fractional system has hereditary properties. Hence, as part of understanding the dynamic behaviour, in this paper, we will perform the computation of stability criterion for a fractional Shimizu–Morioka system. Different from the existing stability analysis for a fractional dynamical system in literature, we apply the optimal Routh–Hurwitz conditions for this fractional Shimizu–Morioka system. Furthermore, we introduce the way to calculate the range of adjustable control parameter β to obtain the stability criterion for fractional Shimizu–Morioka system. The result will be veriﬁed by using the predictor-corrector scheme to obtain the time series solution for the fractional Shimizu–Morioka system. The ﬁndings of this study can provide a better understanding of how adjustable control parameter β inﬂuences the stability criterion for fractional Shimizu–Morioka system.


Introduction
This paper emphasizes the computation of stability criterion for a Shimizu-Morioka system in fractional order as shown below : where σ and β are positive values. Here, we assume β as the adjustable control parameter. The integer order system of Equation (1) was first proposed in [1] and has attracted the interest of researchers to study the stability and various types of bifurcation such as in [2][3][4][5]. However, the fractional order or arbitrary order of the system as in Equation (1) have received less attention; also see [6,7]. Unlike both [6,7], here, we apply optimal fractional order Routh-Hurwitz stability conditions which have recently been derived byČermák and Nechvátal in [8]. Unlike the previous version of Routh-Hurwitz stability conditions for fractional systems (as derived in [9]), these new optimal Routh-Hurwitz condition serve as necessary and sufficient conditions to guarantee that all roots of the characteristic polynomial obtained from the linearization process are located inside the Matignon stability sector. Furthermore, the optimal Routh-Hurwitz conditions obtained were in an explicit form. The optimal Routh-Hurwitz conditions were successfully applied to study the fractional dynamical systems such as

Integer Order Shimizu-Morioka System
The integer order Shimizu-Morioka system as shown in Equation (2).
where σ and β are positive values. Here, we assume β as the adjustable control parameter. If β > 0, then two equilibria appear, which are E ± = (± β, 0, 1). Then, we obtain the general characteristics equation of the system from the linearization along the equilibria.
and by substituting E ± into Equation (3), we have the characteristics equation as follow, From the standard Routh-Hurwitz conditions, it is easy to show that that all the λ in the Equation (4) have negative real parts if and only if In other words, the integer order Shimizu-Morioka system is stable if and only if 0 < β < β * = 2−σ 2 σ .

Fractional Order Shimizu-Morioka System
Throughout this paper, we consider the standard fractional Shimizu-Morioka system as follows: where C 0 D α t is the Caputo derivative operator defined as in Definition 1.

Optimal Routh-Hurwitz Conditions for Fractional System
In the classical theory of Routh-Hurwitz conditions, for three dimensional dynamical system, the characteristic polynomial in cubic form when α = 1 is as in Equation (8) P(λ; a, b, c) = λ 3 + aλ 2 + bλ + c.
Let us consider a, b, c are real coefficients. The Routh-Hurwitz conditions which for the characteristic polynomial in Equation (8) have all zeroes, i.e., λ with negative real parts if and only if a > 0, b > 0 and 0 < c < ab.
Then, the Routh-Hurwitz conditions are only sufficient for validity of the Magtinon stability sector in Equation (10) with 0 < α < 1. Under this situation, the stability sector of the set of all triplets (a, b, c) is wider and significantly more complicated due to Equation (10).
Follow the work in [8], we define BL(α) as the boundary locus as follow.

Stability Analysis of the Fractional Shimizu-Morioka System
In this section, we will compute the stability criterion for the fractional Shimizu-Morioka system as in Equation (6). From Section 2, we will limit our value of σ in [0, . From the calculation, the coefficients of characteristic polynomial as in Equation (8) for fractional system in Equation (6) are a = β + σ > 0, b = βσ > 0 and c = 2β > 0. Thus we can analyze local asymptotic stability of the equilibria by applying fractional optimal Routh-Hurwitz conditions [8]. It is obvious that the system in Equation (6) are locally asymptotically stable for any β > 0 if 0 < α ≤ 2/3. If 2/3 < α < 1, then Theorem 1(i) in Section 3 will be applied. In this case, since all the a, b and c for the characteristic polynomial are bigger than 0, the only relevant condition for calculating the stability criterion is the inequality c < c − (a, b; α) . Hence, we shall explain some basic steps of the computational procedures to obtain the stability criterion for this fractional Shimizu-Morioka system.
By substituting the a = β + σ > 0, b = βσ > 0 and c = 2β > 0 of the fractional Shimizu-Morioka system into c < c − (a, b; α), we can get the inequality as follows We should notice that when α is approaching to 1, the inequality in Equation (16) becomes Equation (5) actually. This can be said that this computation is the generalization of integer order system to fractional order or arbitrary order system. Now, we can analyze the inequality Equation (16) by squaring it and reducing it into polynomial, Q(β).
If we fixed the value of σ, we can evaluate the dependence of abjustable control parameter β on the α, i.e., fractional derivative order in the Caputo sense. In this case, from the result in Section 2, we shall limit the σ only from 0 to √ 2. Thus, we illustrate the fractional Shimizu-Morioka system as in Equation (6) with the value σ = 0.736. We obtain the following fractional system.
We will further explain the above theorems in following section.

Numerical Results
In this section, we will present some simulation of the stability analysis for the system Equation (1). For the simulation purpose, we applying generalized Adams-Bashforth-Moulton type predictor-corrector scheme for solving fractional differential equations developed in [20]. This numerical approach is widely used in the study of the dynamical behaviour of fractional dynamical system, such as in [21,22]. This is because most of the system of fractional differential equation or fractional dynamical system do not have analytical solution. Furthermore, we had also modified this approach for solving differential equation in Caputo-Fabrizio sense as in [23].
From the result in Section 4, when σ = 0.736, we will obtain β = 0.1827310163 and the critical value of α is 0.9001093006. This condition is the fractional order Hopf bifurcation and where periodic solution should be occurred. Perturbate the adjustable control parameter β which leads to the dissolve of limit cycle to unstable or stable condition. For the sake of simplicity when doing the simulation, we take α = 0.9, β = 0.1827, σ = 0.736 and the initial condition as x(0) = 0.5, y(0) = 0, z(0) = 0.8. Figure 1 shows the periodic solution for the time series solution while the phase portrait of the trajectories with the initial condition [0.5, 0, 0.8] shows the limit cycle appeared.  When increasing the value of adjustable control parameter β, the solution becomes to stable toward the equilibrium ( β, 0, 1). We present the simulation result when β = 1 as in Figure 2.
On the other hand, based on Equation (16), the inequality is given as following.
We can reach the conclusion for the computation of stability criterion for fractional Shimizu-Morioka system by using optimal Routh-Hurwitz conditions as in Table 1 which follow Theorems 2 and 3.
If α > α cr , solving the inequality in Equation (30) will give the range of β shows the fractional system is stable. As example, for the case when σ = 0.736, where α cr = 0.9001093006. If we let α = 0.95 which is > α cr , and we substitute it into Equation (30), we will be able to get the range of β which give stability condition for the system. Following Theorem 3, we get β * = 0.01114995609 and β * * = 1.025145474. For other values of α, the detail of calculations is shown in Table 2.   For illustration purpose, we take α = 0.95, the range of β must be [0, 0.01114995609], [1.025145474, ∞] in order to get the stable solution. If the range of β do not lie in that interval, the equlibria will be locally asymptotically not stable. We verified it through simulation for α = 0.95, β = 0.1827, σ = 0.736 as in Figure 3. All the calculations have done by using Maple.

Conclusions
In this paper, we have used new optimal Routh-Hurwitz conditions which also serve as necessary and sufficient conditions to compute the stability criterion for a fractional Shimizu-Morioka system. Here, we summarise our work as the following: • For the σ = [0, √ 2], the new optimal Routh-Hurwitz conditions enable us to detect the critical value of α for the stability criterion of fractional Shimizu-Morioka system when we use β as control parameter.
• Furthermore, we introduce the way to calculate the range of adjustable control parameter β to obtain the stability criterion for the fractional Shimizu-Morioka system.
The result was verified by the famous predictor-corrector scheme for fractional systems. The advantage of using this new optimal Routh-Hurwitz condition is that it enables us to obtain the range of β which fulfills the stability criterion for the fractional Shimizu-Morioka system. However, lot of work need to be done by applying this new optimal Routh-Hurwitz condition to other fractional systems and this will probably be part of our future research work.