Stability of Gene Regulatory Networks Modeled by Generalized Proportional Caputo Fractional Differential Equations

Abstract

A model of gene regulatory networks with generalized proportional Caputo fractional derivatives is set up, and stability properties are studied. Initially, some properties of absolute value Lyapunov functions and quadratic Lyapunov functions are discussed, and also, their application to fractional order systems and the advantage of quadratic functions are pointed out. The equilibrium of the generalized proportional Caputo fractional model and its generalized exponential stability are defined, and sufficient conditions for the generalized exponential stability and asymptotic stability of the equilibrium are obtained. As a special case, the stability of the equilibrium of the Caputo fractional model is discussed. Several examples are provided to illustrate our theoretical results and the influence of the type of fractional derivative on the stability behavior of the equilibrium.

Minimization Problems for Functionals Depending on Generalized Proportional Fractional Derivatives

“…Despite being a very recent idea, we have already found numerous works on this subject. For example, in [20][21][22] we find some fundamental properties of it, stability of fractional differential equations were addressed in [23][24][25], in [26] is studied stochastic differential equations, and a more general form of the derivative, with dependence of an arbitrary kernel, was considered in [27][28][29]. However, with regard to the calculus of variations, no study has yet been carried out and with this work, we intend to contribute to this area.…”

Boundary Value Problem for Impulsive Delay Fractional Differential Equations with Several Generalized Proportional Caputo Fractional Derivatives

“…[6,7] for derivatives involving arbitrary kernels). In the last decade, the generalized proportional fractional derivative is defined [8,9] and studied (see, for example, for stability properties [10][11][12] and for stochastic fractional differential equations [13]).…”

A generalized proportional Caputo fractional model of multi-agent linear dynamic systems via impulsive control protocol

“…Motivated by this research, we consider multi-agent linear dynamic systems with the generalized proportional Caputo fractional derivative and an impulsive control protocol. The generalized proportional Caputo fractional derivative was introduced in [34] and subsequently studied in [35][36][37][38] as an undeviating generalization of the existing Caputo fractional derivative. Namely, in this derivative, we have two parameters: α ≥ 0 which is the order of the derivative, and ρ ∈ (0, 1] which could be called the proportionality parameter.…”