No 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.

<abstract><p>In this paper, we derive the explicit analytical solution of incommensurate fractional differential equation systems with fractional order $ 1 &lt; \alpha, \beta &lt; 2 $. The derivation is extended from a recently published paper by Huseynov et al. in ^{[<xref ref-type="bibr" rid="b1">1</xref>]}, which is limited for incommensurate fractional order $ 0 &lt; \alpha, \beta &lt; 1 $. The incommensurate fractional differential equation systems were first converted to Volterra integral equations. Then, the Mittag-Leffler function and Picard's successive approximations were used to obtain the analytical solution of incommensurate fractional order systems with $ 1 &lt; \alpha, \beta &lt; 2 $. The solution will be simplified via some combinatorial concepts and bivariate Mittag-Leffler function. Some special cases will be discussed, while some examples will be given at the end of this paper.</p></abstract>

Atlantic ocean thermohaline circulation is a deep ocean circulation occur in the Atlantic ocean which shows mixed of salt and freshwater transportation. The ocean circulation box model is defined to cover the large-scale behavior of the thermohaline circulation. On the other hand, fractional order dynamical systems are more flexible and realistic for real-life problems if compare with integer order dynamical systems. Hence, research on the stability for fractional dynamical systems is still infant and more difficult to analyze analytically. In this paper, we will extend the ocean circulation 3-box model into fractional order and investigate stability criterion for this fractional model by applying fractional Routh-Hurwitz conditions. Routh-Hurwitz conditions allow us to find the range of adjustable control parameter F1 which can detect the stability criterion for the fractional ocean circulation model.