The Qualitative Analysis Of A Class Of Cubic Systems

Differential equation is one of the most important applications for differential and integral calculus in the field of mathematical physics research, it has been de-veloping rapidly in the19th century, and emerge a series of research theory which is of great significance. In the end of the19th century, real field differential equa-tion qualitative theory which is founded by Poincare is one of the most important theoretical results. By applying the classic method of qualitative theory of differ-ential equations, three kinds of cubic systems are discussed in this paper under the conditions of different parameters. This article consists of four parts.In the first chapter, we firstly introduce the research background of differential equation; secondly, introducing the main work of this article and some preliminary knowledge which is useful in the paper.In the second chapter, with Lyapunov formal series, stability of index theorem and Hopf bifurcation theory the qualitative nature of the E31system under the first condition are discussed.In the third chapter, by using geometric theory of differential equations and the uniqueness theorem of generalized Lienard system, some sufficient conditions for the uniqueness of limit cycles of the E31system under the second condition are established.In the forth chapter, we study the E31system under the third condition, and obtain some sufficient conditions on the uniqueness of limit cycles of the one by applying qualitative theory of differential equations and the uniqueness theorem of generalized Lienard system.