Abstract
This paper presents a modified chaotic system under the fractional operator with singularity. The aim of the present subject will be to focus on the influence of the new model’s parameters and its fractional order using the bifurcation diagrams and the Lyapunov exponents. The new fractional model will generate chaotic behaviors. The Lyapunov exponents’ theories in fractional context will be used for the characterization of the chaotic behaviors. In a fractional context, the phase portraits will be obtained with a predictor-corrector numerical scheme method. The details of the numerical scheme will be presented in this paper. The numerical scheme will be used to analyze all the properties addressed in this present paper. The Matignon criterion will also play a fundamental role in the local stability of the presented model’s equilibrium points. We will find a threshold under which the stability will be removed and the chaotic and hyperchaotic behaviors will be generated. An adaptative control will be proposed to correct the instability of the equilibrium points of the model. Sensitive to the initial conditions, we will analyze the influence of the initial conditions on our fractional chaotic system. The coexisting attractors will also be provided for illustrations of the influence of the initial conditions.
Highlights
In the recent years, modeling chaotic and hyperchaotic systems occupy an important place in the literature and have many applications in physics, biology, electrical circuits, and many other fields [1,2,3,4]. e most used fields for the applications of chaos are modeling electrical circuits, and there exist many papers related to the implementation of the chaotic systems in this domain
There appear many tools for analyzing the chaotic systems as the phase portraits of the system using the numerical discretizations, the bifurcation diagrams to understand the influence of the models’ parameters on the dynamics of the chaotic models, and the Lyapunov exponents used to determine the nature of the chaos. ere exists some chaos as chaotic behaviors and hyperchaotic behaviors
For the advancement of fractional calculus and its application, the readers can refer to the following papers: in [13], the authors address a new numerical scheme for solving fractional differential equations described by Journal of Mathematics
Summary
In the recent years, modeling chaotic and hyperchaotic systems occupy an important place in the literature and have many applications in physics, biology, electrical circuits, and many other fields [1,2,3,4]. e most used fields for the applications of chaos are modeling electrical circuits, and there exist many papers related to the implementation of the chaotic systems in this domain. E proof of this assumption will be subject to further investigations in the future Another contribution addressed in this paper is related to the local stability of the fractional chaotic system’s equilibrium points. We propose feedback control to stabilize the fractional error system after combining the slave chaotic system and the master’s chaotic systems Another contribution of the present paper is that we provide the coexisting attractors for specific values of the model’s parameters at two different initial conditions.
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