Modeling attractors of chaotic dynamical systems with fractal–fractional operators

By Abdon Atangana, Sania Qureshi. Chaos, Solitons & Fractals, 2019

361 citations recorded by the source corpus.

Abstract In this paper, newly proposed differential and integral operators called the fractal–fractional derivatives and integrals have been used to predict chaotic behavior of some attractors from applied mathematics. These new differential operators are convolution of fractal derivative with power law, exponential decay law with Delta-Dirac property and the generalized Mittag-Leffler function with Delta-Dirac property, respectively. These new operators will be able to capture self-similarities in the chaotic attractors under investigation. We presented, in general, three numerical schemes that can be used to solve such systems of nonlinear differential equations. The general conditions for the existence and the uniqueness of the exact solutions are obtained. These new differential and integral operators applied in five selected chaotic attractors with numerical simulations for varying values of fractional order α and the fractal dimension τ have yielded very interesting results. It is to be believed that this research study will open new doors for in-depth investigation in dynamical systems of varying nature.

Subfields: Modeling and Simulation, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/f95a32dcbba8647ef0769b69e285406e1478da0b

Citations in the corpus

• On the solution of a boundary value problem associated with a fractional differential equation (2020, neutral): “In the recent years, a number of problems in physics, engineering, and other sciences have been modeled by means of fractional differential equations; see previous studies 1-10 and references therein.”
• Mathematical analysis of memristor through fractal‐fractional differential operators: A numerical study (2020, neutral): “Atangana and Qureshi [17] captured self-similarities in the chaotic attractors based on the basis of three numerical schemes for systems of nonlinear differential equations.”