Limit cycles and homoclinic networks in two-dimensional polynomial systems.

In this paper, the properties of equilibriums in planar polynomial dynamical systems are studied. The homoclinic networks of sources, sinks, and saddles in self-univariate polynomial systems are discussed, and the numbers of sources, sinks, and saddles are determined through a theorem, and the first integral manifolds are determined. The corresponding proof of the theorem is completed, and a few illustrations of networks for source, sinks, and saddles are presented for a better understanding of the homoclinic networks. Such homoclinic networks are without any centers even if the networks are separated by the homoclinic orbits. The homoclinic networks of positive and negative saddles with clockwise and counterclockwise limit cycles in crossing-univariate polynomial systems are studied secondly, and the numbers of saddles and centers are determined through a theorem, and the first integral manifolds are determined through polynomial functions. The corresponding proof of the theorem is given, and a few illustrations of networks of saddles and centers are given to show the corresponding geometric structures. Such homoclinic networks of saddles and centers are without any sources and sinks. Since the maximum equilibriums for such two types of planar polynomial systems with the same degrees are discussed, the maximum centers and saddles should be obtained, and maximum sinks, sources, and saddles should be achieved. This paper may provide a different way to determine limit cycles in the Hilbert 16th problem.