Research On Markus-Yamabe Conjecture And Boundary Of The Attraction Basin Of Piecewise Smooth Systems

Piecewise smooth dynamical systems are of theoretical significance and have broad applications,which are widely used in engineering science including control field.This thesis mainly contains two aspects of problems of piecewise smooth dynamical systems.The first aspect focuses on problems of the Markus-Yamabe conjecture for planar piecewise linear systems.The Markus-Yamabe conjecture of smooth vector field has been proved to be true for two-dimensional case.It is worth further discussing that whether the conjecture is true if the condition is reduced from smooth to piecewise smooth and what the specific conditions are for the conjecture to be true.The first part is devoted to studying these problems.The second aspect concentrates on problems of boundary of the attraction basin of the origin for saturation control systems.For three-dimensional anti-stable control systems with saturated linear state feedback,as a class of special and very typical piecewise linear systems,the geometric structure of the attraction basin of the origin is complicated and very difficult to study.There are no systematic theoretical results.The second part is devoted to discussing the related problems.The main results of this thesis are as follows:(1)Discontinuous piecewise linear systems with two zones separated by a straight line containing a boundary singularity,in which every subsystem is asymptotically stable are investigated.The explicit parameter conditions for the existence of limit cycles and for the Markus-Yamabe conjecture to be true are obtained.In addition,for continuous piecewise linear systems with two zones separated by a straight line,in which every subsystem is asymptotically stable,the Markus-Yamabe conjecture is proved to be true.This means that this class of systems has a global asymptotically stable equilibrium point.(2)The case that the discontinuous piecewise linear systems with the only separated line containing no singularities are further studied.The similar results are obtained and more different counterexamples to the Markus-Yamabe conjecture are presented.In particular,for continuous planar piecewise linear systems with n+1 zones separated by n parallel straight lines in phase space,in which each of subsystems is asymptotically stable,it is proved that this system has a globally asymptotically stable equilibrium point,therefore the MarkusYamabe conjecture still holds.(3)A class of piecewise smooth systems arising from three-dimensional anti-stable linear system with saturated linear state feedback is investigated.For this class of systems,the equilibrium points other than the origin lie on the boundary of the attraction basin of the origin is proved.This gives strong evidence that the boundary of the attraction basin is homeomorphic to a sphere.In addition,some necessary conditions for the homeomorphism of the boundary of the attraction basin and the sphere are given.