On the Influence of Fractional Derivative on Chaos Control of a New Fractional-Order Hyperchaotic System

On the Influence of Fractional Derivative on Chaos Control of a New Fractional-Order Hyperchaotic System

Ahmed Ezzat Mohamed Matouk (Majmaah University, Saudi Arabia)
DOI: 10.4018/978-1-7998-3122-8.ch005

Abstract

The non-local fractional differential operators have potential applications in many fields of science and technology but especially in the field of dynamical systems. This chapter introduces a new hyperchaotic dynamical system involving non-local fractional differential operator with singular kernel (the Caputo type). The system involves three quadratic nonlinearities and also three equilibrium points. Existence of chaotic and hyperchaotic attractors has been illustrated. Based on Matouk's stability theory of four-dimensional fractional-order systems, the influence of the fractional differential operator on stabilizing the proposed system to its three steady states has been shown. Numerical results have been provided to verify the theoretical analysis. This kind of study is expected to add useful applications to chaos-based secure communications and text encryption.
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Introduction

Fractional calculus (FC) has potential applications to science and engineering; For example it has applications to physics (El-Sayed, 1996; Hilfer, 2000; Khan & Kumar, 2018; Saqib et al, 2018a-d, Khan & Kumar, 2019a; Al-Khedhairi et al, 2019a; Al-Khedhairi et al, 2019b); applications to economic (Laskin, 2000; Chen, 2008; Chen et al, 2011; Zhen et al, 2011; Hegazi et al, 2013b; Khan & Kumar, 2016); applications to social systems (Ali, 2019); applications to mathematical biology (Ahmed & Elgazzar, 2007; Matouk et al, 2015; Elsadany & Matouk, 2015; Matouk et al, 2015; Matouk & Elsadany, 2016; Selvam et al, 2017; Selvam et al, 2018a-b; Ameen & Novati, 2017; Al-Khedhairi et al, 2018) and applications to engineering (Sun et al, 1984; Petras, 2009; Matouk, 2009a; Matouk, 2009b; Matouk 2010; Matouk, 2011a; Matouk, 2011b; Hegazi & Matouk, 2011; Hegazi et al, 2011; Hegazi & Matouk, 2013; Hegazi et al, 2013a; Matouk & Elsadany, 2014; Banu & Nasir, 2015; Nasir & Singh, 2015; Matouk, 2016; El-Sayed et al, 2016a; El-Sayed et al, 2016b; Khan & Kumar, 2019b). On the other hand, FC has also important investigations in some interdisciplinary topics of applied mathematics (Matignon, 1996; Ben Adda F, 1997; Tavazoei et al, 2009; Tavazoei, 2010; Abdel Latif, 2011; El-Sayed & Salman, 2013; Latif et al, 2013; Abdel Kader et al, 2017; Abdel Kader et al, 2018; Abdel Latif et al, 2018; Ali & Ameen, 2019; Saqib et al, 2019a-b; Taghavian & Tavazoei, 2017; Taghavian & Tavazoei, 2018; Taghavian & Tavazoei, 2019).

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