The Inverse Integrating Factors And Integrability Of Degenerate Analytical Systems

The integrability is one of important subjects in the research field of differential equations and dynamical systems.The inverse integral factor is one of important tools studying center problem,integrability problem,and the number of limit cycles and the distribution for a planar polynomial systems.In general,the expression of the inverse integral factor is simpler than the expressions of the first integrals,its domain of definition is usually larger than the domain of definition of the first integral,so how to find the inverse integral factor of a given system plays an important role in determining the qualitative properties of this system.However,for a given system,it is very difficult to determine whether it has an inverse integral factor and is found.The known studies have given the form of inverse integral factors for some special systems.The known studies have given the existence condition of polynomial inverse integral factor for a class of planar polynomial systems defined by the sum of two homogeneous polynomial systems.The first work of this thesis generalizes the above results to the situation of quasi-homogeneous polynomial systems: there always exists a simple inverse integrating factor for a planar quasi-homogeneous polynomial system and its expression is given;and then a sufficient condition that there exist a polynomial inverse integrating factor for a class of planar polynomial systems defined by the sum of two planar quasi-homogeneous polynomial differential system is proven,and through assuming the polynomial inverse integrating factor has a special formal,the explicit expression of such a polynomial inverse integrating factor is constructed,then from these some formulas of polynomial inverse integrating factors are given for some special planar polynomial systems and semi-quasi-homogeneous polynomial systems,finally two examples show that our results generalize the corresponding known conclusions.The second work in this thesis is to study the existence of polynomial inverse integrating factors of standing wave degenerate systems or traveling wave degenerate systems for a class of fourth-degree planar polynomial degenerate system with application background(coming from the Fisher Equation by the Cole-Hopf transformation,traveling wave transformation and time transformation): firstly,a sufficient condition that there exists a polynomial inverse integrating factor for the standing wave degenerate system is obtained using the corresponding results coming from the second chapter;secondly,this kind of traveling wave system has not any polynomial inverse integral factor is proven by using a necessary condition for the existence of the polynomial inverse integral factor.Finally,we make a brief summary and present some problems for future works.