分数阶Willis环脑动脉瘤系统的混沌动力学分析与控制∗

摘要

整数阶Willis环脑动脉瘤系统在描述表现出黏弹性的血液在具有繁杂弹性的血管系统内的复杂血流动力学机理上有一定局限性；鉴于此，本文利用分数阶Caputo微分及其理论，提出分数阶Willis环脑动脉瘤模型(FWAS)：证明FWAS解的存在惟一性；利用相图和Poincaré截面证明FWAS具有混沌特性，是其整数阶形式的合理推广；结合分岔图和倍周期分岔讨论脉冲压、系统阶次对FWAS的影响；采用通过非自治非线性系统的稳定性条件设计合理的控制器，以药物激励项函数作为脉冲函数进行脉冲控制这两种方法，对FWAS进行有效的控制。本文对FWAS的探讨将对脑动脉瘤的研究具有一定的理论指导意义。%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 Poincaré 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.