Plane Degradation System Of The Central Issues

It is well known that the problem to characterize when a singular point of a givensystem is of focus-centre type and then to study its stability is always a subjectsin the qualitative theory of planar analytic di?erential systems.If the singular pointhas a zero linear part and contains some characteristic directions,the problem tocharacterize when a singular point is of focus-centre type and to study its stabilityproblem is still an open problem.This dissertation consists of five chapters .In this dissertation,we study the relation between the characteristic directionand the special direction and then to study the problem of focus-centre type andstability about these systems which are defined by the sum of two homogeneousvector fields as follows.This dissertation consists of five chapters .In the first chapter, the background, significance and main results for the prob-lem to characterize when a singular point of a given system is of focus-centre typeare given.In the second chapter,we present some basic concepts and some known resultsin the qualitative theory of planar analytic di?erential systems.In the third chapter, a general criterion which some direction for an isolatedsingular point of a planar system is a special direction is given.Meanwhile,we alsodiscuss the relation between the characteristic direction and the special direction foran isolated singular point of a planar system.In the fourth chapter,we discuss the condition of focus-centre problem for thesystems which are defined to be the sum of two homogeneous vector fields.In the fifth chapter,we discuss the condition of stability problem for the systemswhich are defined to be the sum of two homogeneous vector fields.An the end of the dissertation,we put forward a problem which needs to havefurther study.