Strange chaotic attractors under fractal-fractional operators using newly proposed numerical methods

Abstract.

Two classes of differentiation and integral operators have been successfully used in modeling real world problems. On the one hand, researchers used the concept of non-local differentiation to capture the heterogeneity of nature linked to heavy-tailed, and many other, nonlocalities’ dependency. On the other hand, they used a local operator with power law setting to capture processes with similarities. Both operators were used for different purposes; however, we shall note that there exist in nature chaotic processes that exhibit both kinds of behavior. Thus, neither the nonlocal operator nor the differential operator with power law setting can capture such processes. In connection to this, a new class of differential operators has been recently introduced. The differential operator has two orders: the first is the fractional order and the second is the fractal dimension. To open new doors to capturing more chaotic behaviors, in this paper, the Thomas cyclically symmetric attractor, the Shilnikov attractor, the Rössler attractor, the Langford attractor and the King Cobra attractor under the domain of newly proposed fractal-fractional derivative operators, are investigated. Three different types of fractal-fractional derivative operators called the Caputo, Caputo-Fabrizio-Caputo and the Atangana-Baleanu-Caputo operators are employed to comprehend the dynamics of the attractors with the help of varying fractional-order $\Omega$ and fractal-dimension $\Delta$ parameters, where $\Omega,\Delta\in ]0, 1]$. The new strange behaviour of the attractors obtained in this study was not possible earlier in the literature for either classical or fractional cases. Various new three-dimensional graphs are presented for the numerical simulations, which have been carried out with the application of well-convergent methods devised to find approximate solutions for the fractal-fractional order systems.