Stability and dynamics of neutral and integro-differential regularized Prabhakar fractional differential systems
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Stability and dynamics of neutral and integro-differential regularized Prabhakar fractional differential systems Shiva Eshaghi1
· Reza Khoshsiar Ghaziani1 · Alireza Ansari1
Received: 16 March 2020 / Revised: 9 July 2020 / Accepted: 7 August 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this work, we investigate the asymptotic stability analysis for two classes of nonlinear fractional systems with the regularized Prabhakar derivative. The stability analysis of the neutral and integro-differential nonlinear fractional systems are studied by assessing the eigenvalues of associated matrix and applying conditions on the nonlinear part of these types of systems. We use a numerical method to solve the fractional differential equations with the regularized Prabhakar fractional derivative. We further present the numerical simulations on several test cases to examine and reveal the complex dynamics of the analytical obtained results. Keywords Asymptotic stability · Regularized Prabhakar fractional derivative · Neutral and integro-differential systems · Generalized Mittag–Leffler function Mathematics Subject Classification 62P05; 26A33
1 Introduction In the last decades, it has been proven through various research that the fractional order systems have advantages over the integer-order counterparts. The derivatives and integrals of arbitrary order are very convenient for describing properties of many real-world physical systems, and the new fractional models are more satisfying than former integer-order ones (Kilbas et al. 2006; Podlubny 1999; Samko et al. 1993). In this sense, with the growing developments in the various fields of science and engineering (Debnath 2003; Eshaghi and
Communicated by José Tenreiro Machado.
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Alireza Ansari [email protected] Shiva Eshaghi [email protected] Reza Khoshsiar Ghaziani [email protected]
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Department of Applied Mathematics, Faculty of Mathematical Sciences, Shahrekord University, P.O.Box 115, Shahrekord, Iran 0123456789().: V,-vol
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Ansari 2016, 2017; Eshaghi et al. 2019, 2020; Esmaeelzade et al. 2020; Golbabai et al. 2019; Mainardi 1994, 1997; Metzler et al. 1995; Nikan et al. 2020a, b), the concepts of stability analysis of the fractional differential systems have attracted increasing interest for many researchers. For example, some authors studied the stability of fractional order nonlinear systems with the Caputo derivative by using the Lyapunov direct method with the concept of the Mittag–Leffler stability (Li and Chen 2010; Liu et al. 2016; Zhang et al. 2011). Priyadharsini analyzed the stability of the nonlinear fractional dynamical systems (the neutral and integro-differential fractional systems) with the Caputo derivative by considering the associated conditions on the nonlinear terms (Priyadharsini 2016). Eshaghi et al. (2020) investigated the stability and chaotic behaviors of a Caputo fractional order system with the chaos entanglement function. Some authors studi
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