| Dynamics theory originated from study of ordinary differential equations has been developing prosperously since half a century before. With the great progress of stable systems, more attention is paid to study of unstable systems such as existence and disappearance of period orbits, homoclinic orbits and chaos.Besides subharmonic bifurcation, some bifurcation theories are based on the monotonicity of period function. So, people have been focusing on occurrence and distributing of critical period, especially on how many local critical periods appear, which is called local critical period bifurcation to be studied in this paper. The corresponding center is called weak center or isochrone defined by Chicone and Jacobs.For two degree polynomial differential system, cubic homogeneous system, cubic reversible system whose corresponding quadratic system has isochrone and other special polynomial differential systems, many conclusions about local critical period bifurcation have been done. However, there is no result for general cubic system. Actually, determining the order of weak center is related with finding originatenator of the idea of polynomial ring, which is very difficult.In this paper, basic theory of local critical period bifurcation is introduced in chapter one, and then in chapter two, the general method to analyze common zeros of polynomials. In chapter three most conclusions about local critical period bifurcation for cubic system are introduced. In chapter four, the method in chapter two is applied to give the order of weak center for a four degree re-versible differential system whose corresponding cubic system has isochrone and to prove that at most five critical periods perturbed from the weak center. The common zeros problem of higher degree polynomials is overcome with techniques of elimination. Moreover, the parameter space is divided in terms of the order of the weak center and the conditions for this system to have an isochrone at the origin are given.
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