Abstract
In this paper, we consider a class of fractional-order systems described by the Caputo derivative. The behaviors of the dynamics of this particular class of fractional-order systems will be proposed and experienced by a numerical scheme to obtain the phase portraits. Before that, we will provide the conditions under which the considered fractional-order system’s solution exists and is unique. The fractional-order impact will be analyzed, and the advantages of the fractional-order derivatives in modeling chaotic systems will be discussed. How the parameters of the model influence the considered fractional-order system will be studied using the Lyapunov exponents. The topological changes of the systems and the detection of the chaotic and hyperchaotic behaviors at the assumed initial conditions and the considered fractional-order systems will also be investigated using the Lyapunov exponents. The investigations related to the Lyapunov exponents in the context of the fractional-order derivative will be the main novelty of this paper. The stability analysis of the model’s equilibrium points has been focused in terms of the Matignon criterion.
Highlights
Basic Fractional Calculus OperatorsLet us recall all the definitions and properties that are essential for our investigations
In this paper, we consider a class of fractional-order systems described by the Caputo derivative. e behaviors of the dynamics of this particular class of fractional-order systems will be proposed and experienced by a numerical scheme to obtain the phase portraits
How the parameters of the model influence the considered fractional-order system will be studied using the Lyapunov exponents. e topological changes of the systems and the detection of the chaotic and hyperchaotic behaviors at the assumed initial conditions and the considered fractional-order systems will be investigated using the Lyapunov exponents. e investigations related to the Lyapunov exponents in the context of the fractional-order derivative will be the main novelty of this paper. e stability analysis of the model’s equilibrium points has been focused in terms of the Matignon criterion
Summary
Let us recall all the definitions and properties that are essential for our investigations. We are interested in the Riemann–Liouville integral, the Caputo derivative, and all properties related to these operators [17, 18]. E following result establishes a relation between the integral representation and the differential system given in (13)–(15). 3. Lu et al’s Fractional-Order System ere exist nowadays many chaotic and hyperchaotic systems, as recalled in Introduction. Many questions appear for modeling chaotic and hyperchaotic systems with fractional operators. Our new objective is to explain concretely what the influence generated by the fractional-order derivative in modeling the above fractional differential systems is. Lu et al.’s fractional model presented has many applications; for example, it can be implemented for electrical circuits. We give a possible example of implementation to prove that Lu et al.’s fractional-order model can be used in real-world modeling problems. e mathematical aspects of equations (13)–(15) are under consideration in this paper
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