This Ph.D.Thesis is devoted to generalized isochronous center conditions or lin- earizable conditions and bifurcations of limit cycles for planar polynomial differential systems. It is composed of five chapters.In Chapter 1, we introduce the historical background and the present progress of problems that concern with centers, integrability, isochronous centers, linearizability and bifurcations of limit cycles for planar polynomial differential systems. The main works of this paper are concluded as well.In Chapter 2, the problem of limit cycles bifurcating from fine foci for a quartic polynomial Z_{3}-equivariant system is investigated. We prove that the system has sixteen small amplitude limit cycles. This is a new result in the study of the second part of the 16th Hilbert problem. The proof of existence of limit cycles is algebraic and symbolic.In Chapter 3, we study quantities at infinity and the appearance of limit cycles from the equator in polynomial vector fields with no singular points at infinity. We start by proving the algebraic equivalence of the corresponding quantities at infinity （also focal values at infinity） for the system and its translational system, then we obtain that the maximum number of limit cycles that can appear at infinity is invariant for the systems by translational transformation. At the same time, a class of cubic polynomial system and its translational system are also investigated, which just exemplifies the above results.In Chapter 4, the complex center and complex isochronous centers for a class of fifth system are discussed. Firstly following the new algorithm to compute period constants to find necessary conditions for complex isochronicity, then applying some effective method to prove the sufficient conditions, actually we obtain the real isochronous centers conditions and the linearizable conditions of the corresponding saddle point.At last Chapter, we give an new algorithm to compute generalized period con- stant. It is not only a method to judge generalized isochronous center, but also a good way to find necessary conditions for linearizable systems for any rational resonance ratio. The algorithm is linear recursive and easy to realized by a recursive function in computer algebra systems. With forcing only addition, subtraction, multiplication and division to the coefficients of the system, the generalized period constants can be deduced. Compared with the known methods, complex integrating calculations and operations of trigonometric functions are avoided in computation. As the application, we discuss linearizable conditions for the Lotka-Volterra systems of 3 : -m resonances ratio and give completely the linearizable conditions of several cases systems. And more linearizability of Hamiltonian system with the origin as 1 : -1 resonances saddle point is also investigated systematically. At the same time, We also develop some technique to prove the sufficient conditions. |