关键词: 分数阶Willis环脑动脉瘤系统/
分数阶Caputo微分/
Poincar/
截面/
混沌控制
English Abstract
Chaotic dynamics of the fractional Willis aneurysm system and its control
Gao Fei,Li Teng,
Tong Heng-Qing,
Ou Zhuo-Ling
1.School of Science, Wuhan University of Technology, Wuhan 430070, China
Fund Project:Project supported by the Major Research plan of the National Natural Science Foundation of China (Grant No. 91324201) and the Natural Science Foundation of Hubei Province, China (Grant No. 2014CFB865).Received Date:05 June 2016
Accepted Date:06 September 2016
Published Online:05 December 2016
Abstract: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.
Keywords: fractional Willis aneurysm system/
Caputo fractional differential/
Poincar/
section/
chaos control
