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Almeida R, Agarwal RP, Hristova S, O’Regan D. Stability of Gene Regulatory Networks Modeled by Generalized Proportional Caputo Fractional Differential Equations. ENTROPY 2022; 24:e24030372. [PMID: 35327883 PMCID: PMC8947342 DOI: 10.3390/e24030372] [Citation(s) in RCA: 5] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 02/07/2022] [Revised: 02/28/2022] [Accepted: 03/04/2022] [Indexed: 01/25/2023]
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.
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Affiliation(s)
- Ricardo Almeida
- Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal;
| | - Ravi P. Agarwal
- Department of Mathematics, Texas A&M University-Kingsville, Kingsville, TX 78363, USA;
| | - Snezhana Hristova
- Faculty of Mathematics and Informatics, Plovdiv University “P. Hilendarski”, 4000 Plovdiv, Bulgaria
- Correspondence:
| | - Donal O’Regan
- School of Mathematical and Statistical Sciences, National University of Ireland, H91 TK33 Galway, Ireland;
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Further results on asymptotic and finite-time stability analysis of fractional-order time-delayed genetic regulatory networks. Neurocomputing 2022. [DOI: 10.1016/j.neucom.2021.11.088] [Citation(s) in RCA: 6] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022]
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Lyapunov Approach for Almost Periodicity in Impulsive Gene Regulatory Networks of Fractional Order with Time-Varying Delays. FRACTAL AND FRACTIONAL 2021. [DOI: 10.3390/fractalfract5040268] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
This paper investigates a class of fractional-order delayed impulsive gene regulatory networks (GRNs). The proposed model is an extension of some existing integer-order GRNs using fractional derivatives of Caputo type. The existence and uniqueness of an almost periodic state of the model are investigated and new criteria are established by the Lyapunov functions approach. The effects of time-varying delays and impulsive perturbations at fixed times on the almost periodicity are considered. In addition, sufficient conditions for the global Mittag–Leffler stability of the almost periodic solutions are proposed. To justify our findings a numerical example is also presented.
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Abstract
In this manuscript, we studied a class of delayed Fuzzy Genetic Regulatory Networks (FGRNs) with Stepanov-like weighted pseudo almost automorphic coefficients. New criteria for the existence, uniqueness and global exponential stability of its weighted pseudo almost automorphic solution are established. Our approach is based on Banach fixed point theorem and novel analysis techniques. Moreover, a numerical example is given to illustrate the validity of the obtained results.
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Affiliation(s)
- Chaouki Aouiti
- University of Carthage, Faculty of Sciences of Bizerta, Department of Mathematics, Research Units of Mathematics and Applications UR13ES47, 7021 Zarzouna, Bizerta, Tunisia
| | - Farah Dridi
- University of Carthage, Faculty of Sciences of Bizerta, Department of Mathematics, Research Units of Mathematics and Applications UR13ES47, 7021 Zarzouna, Bizerta, Tunisia
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