Variable-order Fractional Systems Research Articles

GENAVOS: A New Tool for Modelling and Analyzing Cancer Gene Regulatory Networks Using Delayed Nonlinear Variable Order Fractional System

Gene regulatory networks (GRN) are one of the etiologies associated with cancer. Their dysregulation can be associated with cancer formation and asymmetric cellular functions in cancer stem cells, leading to disease persistence and resistance to treatment. Systems that model the complex dynamics of these networks along with adapting to partially known real omics data are closer to reality and may be useful to understand the mechanisms underlying neoplastic phenomena. In this paper, for the first time, modelling of GRNs is performed using delayed nonlinear variable order fractional (VOF) systems in the state space by a new tool called GENAVOS. Although the tool uses gene expression time series data to identify and optimize system parameters, it also models possible epigenetic signals, and the results show that the nonlinear VOF systems have very good flexibility in adapting to real data. We found that GRNs in cancer cells actually have a larger delay parameter than in normal cells. It is also possible to create weak chaotic, periodic, and quasi-periodic oscillations by changing the parameters. Chaos can be associated with the onset of cancer. Our findings indicate a profound effect of time-varying orders on these networks, which may be related to a type of cellular epigenetic memory. By changing the delay parameter and the variable order functions (possible epigenetics signals) for a normal cell system, its behaviour becomes quite similar to the behaviour of a cancer cell. This work confirms the effective role of the miR-17-92 cluster as an epigenetic factor in the cancer cell cycle.

Sliding mode control for a class of variable-order fractional chaotic systems

This paper concerns the stability analysis and controller design for a class of variable order fractional chaotic systems. By use of Fractional Comparison Principle, the Mittag–Leffler stability criterion is proposed and proved for variable order fractional systems. For the unperturbed system, two different kinds of sliding mode controllers are proposed and the stability of the controlled systems is proved based on the proposed stability criterion. The first control law is designed on the constructing of a variable order fractional integral sliding mode surface and results in a free of chattering signal. While, the second one introduces a variable order fractional derivative sliding mode surface, which is also adapted for the variable order fractional system with uncertainty and external disturbance. And the sign · function in the switching control law is transferred into the fractional derivative of the control input such that it avoids the undesirable chattering. In addition, for the systems with uncertainty and external disturbance, an adaptive sliding mode control law is designed for the variable order fractional chaotic system. And the unknown bounds of the uncertainty and external disturbance are estimated by the variable order fractional derivative adaptive laws. Based on the fractional order Barbalat’s Lemma which is extended from the integer order Barbalat’s Lemma, the asymptotical stability is proved for the controlled uncertain system. At last, numerical simulations are presented to verify the validity and efficiency of the proposed fractional order controllers.

Variable order fractional systems

Fractional Calculus had a remarkable evolution during recent decades, and paved the way towards the definition of variable order derivatives. In the literature we find several different alternative definitions of such operators. This paper presents an overview of the fundamentals of this topic and addresses the questions of finding out which of them are reasonable according to simple criteria used for constant order fractional derivatives. This approach leads to the definitions of variable order fractional derivative based on the Grünwald–Letnikov and the Liouville formulations defined on R, as well as to the definition of a Mittag-Leffler function for variable orders, and to the application of these definitions to dynamical systems.