Chapter 15 - Centers, foci, and limit cycles for polynomial differential systems

Abstract

This chapter reviews some symbolic approaches based on characteristic sets, triangular systems, Grobner bases, resultants, and Sturm-like theorems, and summarizes a number of results achieved by using these approaches for cubic differential systems. Besides these general bounds for general systems, the existence of limit cycles and the related problem of centers and foci have also been studied for some specific systems such as cubic ones. Some problems and results for concrete (cubic) systems are reviewed, and symbolic computation and manipulation for polynomials arising from these problems are emphasized. For different classes of cubic systems, six, seven, and eight small-amplitude limit cycles may be constructed. The relationship among the independently obtained center conditions is explained, and necessary and sufficient conditions for the system to have two centers are also given. New conditions on the existence of eight limit cycles for a class of cubic differential systems were derived by solving a system of large polynomial equations and inequalities. The introduction of more variables may increase the computational complexity sharply, and thus make the method not work for large polynomial inequalities. A survey of the work for bifurcating small-amplitude limit cycles for cubic systems using symbolic computation and computer algebra systems is also elaborated in this chapter.