Computer Algebraic Methods for Investigating Plane Differential Systems of Center and Focus Type

For plane differential systems of center and focus type, the author described an algorithmic procedure based on the principle of Poincare’s method and implemented a program DEMS for computing the Liapunov function and Liapunov constants. This function and these constants are used in the study of stability criteria, differentiation between center and focus and the construction of limit cycles. The solutions of the problems concerning the investigation of Liapunov constants then require that algebraic decision problem, algebraic simplification and algebraic equations solving, to which Wu’s characteristic set method and Buchberger’s Gröbner basis method are successfully applied. Using the program DEMS and these computer algebraic methods, the author studied some concrete differential systems and obtained the stability criteria and the relations between the computed Liapunov constants and other conditions. In particular, we discovered that Kukles’ conditions for the existence of a center for a type of cubic differential systems are possibly incomplete, and presented a class of cubic differential systems with the origin as a 6-tuple focus from which one can create 6 limit cycles by a small perturbation. This paper is a summarization of our recent work.