The Studies Of Several Classes Of Differential Autonomous Systems

In this paper, using qualitative theory of di?erential equation, the problems oflinearizability and limit cycles for several classes of di?erential equations are studied.This thesis is composed of six chapters.In Chapter one, we introduce the historical background of limit cycles and lin-earization for planar polynomial di?erential systems. The main works of this paper areconcluded as well.We introduce some preliminaries in Chapter two.In Chapter three, The linearization of a class of Lotka-Volterra system is investi-gated. First, we get a couple of first linearized value of p: -1 resonance Lotka-Volterrasystem by calculating and deducing, at last, we get the necessary and su?cient lin-earization of the system.In Chapter four, we study the necessary and su?cient linearizability of a class ofquasi-cubic di?erential system. First of all, we transfer the system into quintic complexsystem using specific conversion, and get limited numbers of linearizability quantitiesof the origin of the system, Second, we get the necessary condition after simplifyingthe quantities. At last, using the ways of the Darboux algebraic invariant curve, Liebrackets, integral factor, first integration and so on, we prove those conditions aresu?cient.In Chapter five, we study the quartic Hamiltonian system with quartic perturbedterms using the accurate method and get the result that the system has at least 14limit cycles.In Chapter six, we sum up the whole thesis, explain the unsolved problems andprospect the research in the future.