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

Abstract

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.

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“… 26 The Adams‐type Predictor‐Corrector approach can further be considered to solve other numerical problems such as nonlinear differential equations 27 and Fractional Shimizu–Morioka problems. 28 …”

“…26 The Adams-type Predictor-Corrector approach can further be considered to solve other numerical problems such as nonlinear differential equations 27 and Fractional Shimizu-Morioka problems. 28 To demonstrate the model stability as considered in (4), the value of parameters are taken as follows: The initial values are S p (0) = 32351818; E p (0) = 31927442; I p (0) = 389846; A p (0) ∶= 200; R p (0) ∶= 389846; M(0) ∶= 50000, 𝜃 p = 0.413, 𝜂 w = 0.000001231, 𝜇 p = 0.00500, 𝜂 p = 0.05, Π p = 107644.22451, w p = 0.00047876, 𝜌 p = 0.005, 𝜏 p = 0.09871, 𝜏 ap = 0.3912, Q p = 0.000298, 𝜛 p = 0.0001, 𝜋 = 0.01.…”

“…The method was further studied by El‐Saka 21 and Ameen and Novati, 26 to possess an error‐free approach for obtaining the solution of a problem with a logical and sensible choice of time step 26 . The Adams‐type Predictor‐Corrector approach can further be considered to solve other numerical problems such as nonlinear differential equations 27 and Fractional Shimizu–Morioka problems 28 …”