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## Multi-directional and saturated chaotic attractors with many scrolls for fractional dynamical systems

 Department of Mathematical Sciences, University of South Africa, Florida, 0003, South Africa

* Corresponding author: franckemile2006@yahoo.ca

Received  April 2018 Revised  May 2018 Published  March 2019

Fund Project: This work was partially supported by the grant No: 105932 from the National Research Foundation (NRF) of South Africa

Chaotic dynamical attractors are themselves very captivating in Science and Engineering, but systems with multi-dimensional and saturated chaotic attractors with many scrolls are even more fascinating for their multi-directional features. In this paper, the dynamics of a Caputo three-dimensional saturated system is successfully investigated by means of numerical techniques. The continuity property for the saturated function series involved in the model preludes suitable analytical conditions for existence and stability of the solution to the model. The Haar wavelet numerical method is applied to the saturated system and its convergence is shown thanks to error analysis. Therefore, the performance of numerical approximations clearly reveals that the Caputo model and its general initial conditions display some chaotic features with many directions. Such a chaos shows attractors with many scrolls and many directions. Then, the saturated Caputo system is indeed chaotic in the standard integer case (Caputo derivative order $\alpha = 1$) and this chaos remains in the fractional case ($\alpha = 0.9$). Moreover the dynamics of the system change depending on the parameter $\alpha$, leading to an important observation that the saturated system is likely to be regulated or controlled via such a parameter.

Citation: Emile Franc Doungmo Goufo. Multi-directional and saturated chaotic attractors with many scrolls for fractional dynamical systems. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020034
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Representation of the phase portrait of saturated functions. In (a), the one-variable saturated function (4). In (b), the phase portrait of saturated function series (5) with $k = 1, \ \ h = 4$
Plot representing multi-dimensional simulations of the saturated system (1) with $\alpha = 1$ (conventional case). In (a), projection of a one-dimensional saturated chaotic attractor with six scrolls. In (b), a projection of a two-dimensional saturated chaotic attractor with a grid of $6\times6$ scrolls. In (c), a projection of a three-dimensional saturated chaotic attractor with a grid of $6\times6\times6$ scrolls
Plot representing multi-dimensional simulations of the saturated system (1) with $\alpha = 0.9$ (fractional case). The same dynamics as Fig. 2} are shown with a one-dimensional chaotic attractor with six scrolls in (a), two-dimensional saturated chaotic attractor with a grid of $6\times6$ scrolls in (b) and a three-dimensional saturated chaotic attractor with a grid of $6\times6\times6$ scrolls in (c)
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