Journal
CHAOS SOLITONS & FRACTALS
Volume 123, Issue -, Pages 320-337Publisher
PERGAMON-ELSEVIER SCIENCE LTD
DOI: 10.1016/j.chaos.2019.04.020
Keywords
Fractal-fractional derivatives; Fractal-fractional integral; Self-similar attractors; New numerical schemes
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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 alpha and the fractal dimension tau 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. (C) 2019 Elsevier Ltd. All rights reserved.
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