In the study of Hilbert 16th problem the most difficult part is to find the maximal number of limit cycles appearing near a polycycle by perturbations. In this paper we study the bifurcation of limit cycles near a polycycle with n hyperbolic saddle points. We obtain a sufficient condition for the polycycle to generate at least n limit cycles. We also establish a necessary and sufficient condition for the existence of a separatrix connecting any two saddle points.

In this paper, we develop Kaplan-Yorke's method and consider the existence and bifurcation of 2r2kþ1-periodic solutions for the high-dimensional delay differential systems. We also study the periodic solution and its bifurcation for this system with parameters and present some application examples.

In this paper we develop Kaplan-Yorke's method and consider the existence of periodic solutions for some delaydifferential equations. We especially studyHopf and saddle-node bifurcations of periodic solutions with certain periods for these equations with parameters, and give conditions under which the bifurcations occur. We also give application examples and find that Hopf and saddle-node bifurcations often occur infinitelymanytimes.

In this paper we give a criterion for the stability of planar double homoclinic and heteroclinic cycles with one or two saddles in some degenerate case.

The focus of the paper is mainly on the existence of limit cycles of a planar system with third-degree polynomial functions. A previously developed perturbation technique for computing nor-mal forms of di erential equations is employed to calculate the focus values of the system near equilibrium points. Detailed studies have been provided for a number of cases with certain re-strictions on system parameters, giving rise to a complete classi cation for the local dynamical behavior of the system. In particular, a su cient condition is established for the existence of k small amplitude limit cycles in the neighborhood of a high degenerate critical point. The condi-tion is then used to show that the system can have eight and ten small amplitude (local) limit cycles for a set of particular parameter values.