关键词: 时滞/
分数阶/
Willis环脑动脉血管瘤系统/
混沌控制
English Abstract
Chaotic analysis of fractional Willis delayed aneurysm system
Gao Fei,Hu Dao-Nan,
Tong Heng-Qing,
Wang Chuan-Mei
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), the Fundamental Research Funds for the Central Universities, China (Grant No. 2018IB017), the Natural Science Foundation of Hubei Province, China (Grant No. 2014CFB865), and the Humanity and Social Science Youth foundation of Ministry of Education of China (Grant No. 14YJCZH143).Received Date:02 February 2018
Accepted Date:16 April 2018
Published Online:05 August 2018
Abstract:The dynamic system of Willis aneurysm (WAS) has played an important role in theoretical and clinical research of cerebral aneurysms. Fractional differential is an effective mathematical tool that can describe the cerebral aneurysm system accurately and profoundly. However, the existing fractional Willis aneurysm system (FWAS) cannot describe the delayed aneurysm rupture of unknown cause in reality. Therefore, by introducing the time-delay factors into the existing fractional Willis aneurysm system as a rational extension, a new fractional Willis aneurysm system with time-delay (FWASTD) is proposed in this paper.First, FWASTD is introduced in the context, and the comparison of time sequences map between FWAS and FWASTD proves that FWASTD is feasible in the depiction of time-delay situations. The bifurcation diagram and the largest Lyapunov exponent diagram as well as the phase diagram of fractional order also confirm the chaotic characteristics of the FWASTD.Then, the classical analysis methods in chaotic dynamics, such as time series diagram, phase diagram and Poincar section are used to analyze FWASTD in detail. When studying the diagrams of time-delay factors for the important physiological parameters of the system, we find that blood flow resistance coefficient can exert a remarkable effect on the system stability under time-delay. Besides, the experimental results show that the FWASTD becomes stable with the increase of blood flow resistance under a certain condition. Usually, promoting thrombosis is a kind of adjunctive therapy in clinic for cerebral aneurysm. The results of this part can accord with the treatment in clinic and has great significance in clinical diagnosis.Finally, as the chaotic state of the time-delay system indicates that cerebral aneurysm is in a dangerous situation, the primary task of the control for this new system is to achieve stability rather than synchronization. The stability theory of fractional time-delayed system is adopted in a strict proof of the uniqueness of solution for the FWASTD. To make FWASTD stable, a corresponding linear controller is designed based on the stability theory of fractional order delay system. The numerical simulation indicates that the linear controller can control the blood flow velocity and speed up the periodic fluctuation within a small range, which illustrates that it is not easy to rupture the cerebral aneurysm. We also make self-synchronization control between FWASTD and FWAS just in case that the coefficients of the system are not clear.The research results in this paper, to some extent, can serve as theoretical guidance for the clinical diagnosis and the treatment of aneurysm.
Keywords: time delay/
fraction order/
Willis aneurysm system/
chaos control
