Development and Elaboration of a Compound Structure of Chaotic Attractors with Atangana–Baleanu Operator

Abstract

After the finding of the compound structure for standard chaotic attractors, the main concern was related to how to regulate such a fascinating dynamic. Hence, the question about the existence of a compound structure for chaotic attractors generated by fractional systems was raised. In this work, we investigate the existence of compound structure of a chaotic attractor generated from a Atangana–Baleanu fractional system where two cases are studied: the integer case and the fractional one. The model is first solved numerically thanks to the Haar Wavelets scheme whose convergence is proved via error analysis. Numerical simulations are performed and clearly reveal the existence of the desired compound structure in both cases and characterized by the generation of a left-attractor seen as the reflection of a right attractor through the mirror operation. Moreover, those two simple attractors can always be combined together to form the resulting chaotic attractor. The mechanism of forming those simple attractors is shown and leads to a bounded partial attractor. Furthermore, that same mechanism appears to be strongly dependent on two parameters, the model parameter u and the Atangana–Baleanu derivative with order $\alpha ,$ important in controlling the systems. It is observed that, in the fractional case ($\alpha =0.9$), the period-doubling bifurcations start at a higher value of u compared to the integer case ($\alpha =1$).