Nonlinear Analysis And Control To The System Of Aneurysm Of Circle Of Willis

The systems of aneurysm of circle of Willis are nonlinear systems, which were built up only several years ago and have been developing day after day. They are based on clinical observations and outer-body simulation experiments, and related closely to biomedical meanings. The blood flow speed in aneurysms are taken as the main research object and the development of aneurysms are embodied by the hemodynamics of intracranial aneurysms at the site of circle of Willis .We first analyze the stability of equilibrium points of two Willis systems from the qualitative point of view. All the equilibrium points of the simplified system are unstable and appear bifurcation, which provides theoretical basis for the existence of chaotic solutions.We then discuss the stability of periodic movement for a more general system of aneurysm of circle of Willis. By analyzing the stability of the equivalent autonomous differential equation, we derive the conditions when the original aneurysm equation losts stability of the periodic movement, Such research provides patients with the theoretical basis of preventing periodic half headache.Considering the parameters'uncertainty of the aneurysms system, we apply adaptive fuzzy control methods to control the system to a regulatedstate or an expected orbit by adjusting fuzzy rules, we also give out numerical simulation, which proves that the thoughts of adaptive fuzzy control are effective for this nonlinear system.Finally, we propose a new system of aneurysm of circle of Willis in light of Panetta's thoughts of mathematical modelling. Then we respectively study the existence and uniqueness of optimal control. Theoretically, we show under what conditions the fluctuation is smallest and aneurysms will least rupture.These results are of certain clinical meanings for prevention and treatment intracranial aneurysms, they also give a new research issue to medicine with significant biological and medicine meanings.