| In this paper, we investigate the bifurcation of limit cycles of several planar systems near a center or a focus. Using the method of power series and qualitative analysis, we obtain the Hopf cyclicity of two symmetric Liénard systems, and a global result is also presented. Next we consider the bifurcation of limit cycles of planar near-Hamiltonian systems near nilpotent centers, study the smooth properties of the first Melnikov function and its first coefficients, obtain a new bifurcation theorem, especially for a class of cubic systems we give the center conditions. Finally we deal with a quintic near-Hamiltonian system, and obtain larger number of limit cycles by using the method of stability-changing than that by a general method.In Chapter 1, we introduce some results of the second part of Hilbert's 16th problem and its weakened problem, the bifurcation theory and methods of dynamical systems, and then list our main work.In Chapter 2, we study the maximal number of limit cycles of two types of symmetric polynomial Liénard systems near two singular points of index +1. We make some equivalent transformations to simplify our systems, and using the method of power series, we obtain the Hopf cyclicity of these two systems. For some lower-degree systems, global results are given.In Chapter 3, we consider Hamiltonian systems with nilpotent centers. We first give nilpotent center conditions for cubic systems. Then for general perturbed Hamiltonian systems near nilpotent centers, we investigate the smooth properties of the first Melnikov function in detail and the first coefficients of its expansion, and obtain a new bifurcation theorem. Finally, we give an example as an application.In Chapter 4, we investigate limit cycles for a class of quintic near-Hamiltonian systems near nilpotent centers, which is one case of Chapter 3. By using the blowing-up technique and bifurcation theory, we obtain the number of limit cycles near the center, which is more than that by a known result.
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