The Willis aneurysm system has some limitations in the description of the complex hemodynamic mechanism of blood with viscoelasticity. The fractional calculus has been used to depict some complex and disordered processes in organisms. Thus, we propose a fractional Willis aneurysm system (FWAS) byusing the Caputo fractional differential and its theory in the present article.
Firstly, the existence and uniqueness of solution for FWAS are investigated theoretically. Then, we prove that the FWAS has a chaotic characteristic by analyzing the phase portraits and Poincar section, and it is a rational extension of its integer order form. We investigate the influences of pulse pressure and fractional order on the FWAS by means of bifurcation diagram and period doubling bifurcation. The results show that small changes of pulse pressure and fractional order canlead to a remarkable effect on the motion state of the FWAS.
As the chaotic FWAS indicates that the brain blood flow is unstable, and the cerebral aneurysms are more likely to rupture in a very chaotic velocity field. Therefore we use two methods to control the chaotic FWAS. One is to design a suitable controller based on the stability theorem of fractional nonlinear non-autonomous system, and the other is to use a pulse control by taking the inspirit function of drug as impulse function. The numerical simulations show that the proposed two methods can control the blood flow velocity and speed up the periodic fluctuation within a small range, which shows that the cerebral aneurysm is not easy to rupture.
The results obtained in this paper display that the fractional differential is a feasible method to characterize the Willis aneurysm system. The theoretical results in our article can provide some theoretical guidance for controlling and utilizing the actual FWAS system.