Chaotic analysis of Atangana–Baleanu derivative fractional order Willis aneurysm system**Project supported by the State Key Program of the National Natural Science of China (Grant No. 91324201), the Fundamental Research Funds for the Central Universities of China, the Self–determined and Innovative Research Funds of WUT, China (Grant No. 2018IB017), and the Natural Science Foundation of Hubei Province of China (Grant No.

Abstract

A new Willis aneurysm system is proposed, which contains the Atangana–Baleanu(AB) fractional derivative. we obtain the numerical solution of the Atangana–Baleanu fractional Willis aneurysm system (ABWAS) with the AB fractional integral and the predictor–corrector scheme. Moreover, we research the chaotic properties of ABWAS with phase diagrams and Poincare sections. The different values of pulse pressure and system order are used to evaluate and compare their effects on ABWAS. The simulations verify that the changes of pulse pressure and system order are the significant reason for ABWAS’ states varying from chaotic to steady. In addition, compared with Caputo fractional WAS (FWAS), ABWAS shows less state that is chaotic. Furthermore, the results of bifurcation diagrams of blood flow damping coefficient and reciprocal heart rate show that the blood flow velocity tends to stabilize with the increase of blood flow damping coefficient or reciprocal heart rate, which is consistent with embolization therapy and drug therapy for clinical treatment of cerebral aneurysms. Finally, in view of the fact that ABWAS in chaotic state increases the possibility of rupture of cerebral aneurysms, a reasonable controller is designed to control ABWAS based on the stability theory. Compared with the control results of FWAS by the same method, the results show that the blood flow velocity in the ABWAS system varies in a smaller range. Therefore, the control effect of ABWAS is better and more stable. The new Willis aneurysm system with Atangana–Baleanu fractional derivative provides new information for the further study on treatment and control of brain aneurysms.