Existence Problem Of Inverse Integrating Factors Of Analytical System And Its Application

The inverse integrating factor is often used to study the integrability of planar analytic systems,the number of limit cycles and their distribution in planar polynomial systems,and the center problem.However,it is difficult to determine whether a given differential system has an inverse integrating factor.And for systems with inverse integrating factors,how to find its inverse integrating factor is also a topic worth exploring.Existing research is given that the inverse integrating factor is obtained only for some special forms of differential systems.In addition,the inverse integrating factor is closely related to the symmetry of the differential system,so study of the existence of inverse integrating factor of differential system has important theoretical and practical significance.The singular points of the differential equation can be divided into elementary singular points,nil-potent singular points and linear zero singular points.The existing results have proved that the existence and uniqueness of the inverse integrating factors of the special type of singularities for the focus point,the non-resonant hyperbolic node and the Siegel hyperbolic saddle by the normal form theory of the planar analytical system.The first work in this dissertation is to prove the existence of inverse integral factors of other types of elementary singularities,so as to completely solve the problem of the existence of the inverse integrating factors of elementary singularities.Specifically,by studying the relationship between the inverse integrating factors of the conjugate system and using the normal form of the elementary singular points of the analytic planar system,we study the condition of existence the inverse integrating factors of form V(?(x,y)),and give the specific expression of the inverse integrating factors.And hence the problem of the existence the inverse integrating factors for the elementary singular points of planar systems is completely resolved.The existing result shows that any nil-potent system generally has an algebraic inverse integrating factor in the quotient field C((x,y))of the formal power series algebra C[[x,y]].The second work in this dissertation try to use blow-up technique to extend this result to nonlinear differential system with linear zero part,and in general there is not any algebraic inverse integrating factor over the quotient field of the formal power series algebra.As an example,we also use the blow-up technique to discuss the problem of existence of the inverse integrating of a class of Fisher equations.Finally,we make a brief summary and present some problems for future works.