Problem Of Isochronous Centers Of Planar Differential Systems

A singular point of planar differential systems is called a center if it has a neighbourhood which is covered with periodic orbits. A center is called an isochronous center if those periodic orbits in some neighbourhood have the same period. The investigation of the periods of periodic orbits surrounding a center takes an important role in the study of qualitative theory because there are close connections between the properties of the periods and other qualitative properties. For instance, a nondegenerate condition in the study of heteroclinic and subharmonic bifurcations is that the period is localy strictly monotone. Thus, it is necessary to judge if the concerned center is isochronous (called problem of isochronous centers or Isochronicity of centers). The main difficulties encountered include computing the variety of some polynomials and finding the linearizing transformations or transversal commuting systems. In this thesis, we study the problem of isochronous centers.In recent years, many researchers, such as Chavarriga, Chicone, Christopher, Sabatini and Romanovski, focus on the problem of isochronous centers and many interesting works have been done. However, there is no general method to investigate the problem of isochronous centers and the computation power of computers restricts the development of this problem because many computations are involved. Therefore, the problem of isochronous centers has not been solved for many differential systems including some polynomial differential systems of low degree.In Chapter two, we give a method based on Chouikha's criterion for finding isochronicity conditions for the Lienard-type equation and apply it to the reduced cubic time-reversible system to obtain the isochronicity conditions.In Chapter three, we obtain the necessary conditions for the origin to be an isochronous center of the cubic time-reversible system with the origin being a quadratic isochronous center by computing the variety of some period constants. Then, using the definition of isochronous center or giving the linearizing transformations, we prove the sufficiency of these obtained conditions. Therefore, the conditions for a quadratic isochronous center to keep the isochronicity when the quadratic system has cubic time-reversible perturbations are obtained.In Chapter four, we give the isochronicity conditions for the quintic systems with homogeneous nonlinearities by finding the linearizability conditions for the quintic complex system with homogeneous nonlinearities. Our method is to compute the linearizability quantities and then find the linearizing transformations.In Chapter five, we deduce a sufficient condition for the origin not to be an isochronous center, determine the orders of weak centers and give the maximum number of critical periods bifurcating from the weak centers for the polynomial Hamiltonian systems by computing the period constants implicitly. Applying this result to the Hamiltonian systems which have homogeneous nonlinearities or only have even degree nonlinearities, we prove the non-isochronicity of the center at the origin. Therefore, we answer partly an open problem about isochronous centers given by Jarque and Villadelprat in J. Diff. Equa. in 2002.