Chaotic Analysis And Control Of The Fractional Willis Delayed Aneurysm System

The dynamic system of Willis aneurysm(WAS)has played an important role in theoretical and clinical research of cerebral aneurysms.Fractional differential,as an effective mathematical tool that can describe the soft material and viscoelastic feature of complex fluid such as blood,has been properly used in the WAS for research,which further deepens the mechanism description of WAS.However,the existing fractional Willis aneurysm system(FWAS)cannot explain the delayed aneurysm rupture of unknown cause during the treatment by implanting blood flow guide device and combined vascular stent and coil.Therefore,by introducing the time-delay factors into the existing fractional Willis aneurysm system,a new fractional Willis aneurysm system with time-delay(FWASTD)has been proposed in this paper,aiming at explaining the delayed aneurysm rupture from mathematic model.First,FWASTD is introduced,and the comparison of the time sequences map,phase diagram,Poincaré section of WAS verifies the consistency of depiction between FWASTD,WAS and FWAS both in chaos state and steady state,and shows the validity of extension and depiction system of FWASTD.Then,the comparison of the time sequences map between FWAS and FWASTD proves that FWASTD is feasibility in the depiction of time delay.Last but not least,after comparing some of dynamic classic images of FWAS in some bifurcation joint,it is found that the dynamic behaviour of the system is greatly affected by time-delay.Then,the classical analysis method in chaotic dynamics is used to analyze FWASTD thoroughly,and theoretical research is carried out on the uniqueness of solution in FWASTD.When studying the effects of time-delay on important physiological parameters of the system,it is found that damping coefficient of blood is significant for the system stability under time-delay.Besides,the experimental results show that FWASTD becomes stable with the increase of blood flow resistance under certain condition,and forms relatively specific correspondence with thrombus as adjunctive therapy in clinic.The results of this research has great significance on clinical diagnosis.Finally,based on stability theory of fractional order nonlinear systems with time-delay,a linear controller is designed for FWASTD to demonstrate the controllability of FWASTD.The numerical simulation results shows that the linear controller can control the blood flow speed to fluctuate within a small range,reflected by stability without fracture in aneurysm.This provides a reference for the design of vascular stent in clinic.Meanwhile,two types of synchronization control of FWASTD are achieved.One is synchronization control between FWASTD and FWAS,and the other is self-synchronization control of FWASTD with different coefficients.Also,the simulation has been done to verify its validity.The research in this paper,to some extent,provides feasible theoretical instruction for the clinical diagnosis and treatment of aneurysm.