Abstract
The aim of this article is to analyze the dynamics of the new chaotic system in the sense of two fractional operators, that is, the Caputo–Fabrizio and the Atangana–Baleanu derivatives. Initially, we consider a new chaotic model and present some of the fundamental properties of the model. Then, we apply the Caputo–Fabrizio derivative and implement a numerical procedure to obtain their graphical results. Further, we consider the same model, apply the Atangana–Baleanu operator, and present their analysis. The Atangana–Baleanu model is used further to present a numerical approach for their solutions. We obtain and discuss the graphical results to each operator in details. Furthermore, we give a comparison of both the operators applied on the new chaotic model in the form of various graphical results by considering many values of the fractional-order parameter [Formula: see text]. We show that at the integer case, both the models (in Caputo–Fabrizio sense and the Atangana–Baleanu sense) give the same results.
Highlights
Chaotic models are widely considered due to their applications in many branches of engineering and science and, especially, a rapid increase in the form of publications on chaotic models has been observed.[1,2,3,4,5,6,7] chaotic models are considered for practical purposes in many areas, such as chaotic communication,[2] image watermarking,[1] and autonomous mobile robots.[3]
We considered the numerical scheme presented for the Caputo–Fabrizio derivative model (14) and the Atangana–Baleanu model (23) and obtained the graphical results by taking the fractional-order parameter a = 1, 0:99, 0:97, see Figures 17–25
We presented the dynamics of the new chaotic model in two fractional operators, the Caputo–Fabrizio operator and the Atangana–Baleanu operator
Summary
Chaotic models are widely considered due to their applications in many branches of engineering and science and, especially, a rapid increase in the form of publications on chaotic models has been observed.[1,2,3,4,5,6,7] chaotic models are considered for practical purposes in many areas, such as chaotic communication,[2] image watermarking,[1] and autonomous mobile robots.[3]. The fractional-order modeling has gained a lot of attentions from the researchers once the new operators are defined. In these newly defined operators, the operator Caputo–Fabrizio and the Atangana–Baleanu derivative received more attention from the researchers, and various articles have been proposed on these, including the Caputo derivative.[11,12,13,14,15,16,17,18] There are many applications of the fractional models, such as the generalization of the model and the memory effects, which are usually attached to the real-life problems. The fractional-order modeling has the advantage of observing the dynamics of the real-life problems at any point of interests where such analysis for the integer-order
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