Bi-center Problem And Bifurcation Of Liénard Systems

An isolated equilibrium point of planar vector fields is called a center if it is surrounded by closed orbits in a neighborhood.In 1980s,Poincare asserted that an isolated equilibrium point of planar polynomial vector fields was a center if and only if all focal values of the isolated equilibrium point vanished.Then the investigation of the center problem took an important role in the study of the qualitative theory of plane differential systems.In fact,we can calculate the first several focal values of the equilibrium point but not all.According to the Hilbert Finite Basis Theorem,there must be finitely many focal values to represent all of them.After several decades,a large number of effective methods have been developed to study the center problem.After finding the necessary condition by assuming all the focal values to be zeros,we prove the sufficiency of the condition by finding the formal first integral of the system.Two equilibrium points of planar differential systems are called bi-center if both of them are centers under the same condition.To find the condition to make two equilibrium points to be bi-center is called the bi-center problem.In recent years,many scholars focus on the bi-center problem and many meaningful works have been done.However,it is difficult to decide which system has bi-center.In addition,the development of the bi-center problem is restricted by the computer technology because there are too many algebraic computations involved in calculating focal values.Therefore,the problem of bi-center has not been solved for many differential systems including some cubic polynomial systems.In this thesis,we study the bi-center problem and Hopf bifurcation of some Lienard systems.In Chapter three,we obtain the necessary condition for bi-center of Lienard systems,which have three distinct equilibrium points,with cubic damping term and cubic restore term(the cubic Lienard systems for short).By finding the formal fist integral of the system,we prove the sufficiency of the obtained condition.Therefore,the necessary and sufficient condition of bi-center for the cubic Lienard systems which has three distinct equilibrium points without symmetry are obtained.In Chapter four,we obtain the necessary and sufficient condition of bi-center for the quintic Lienard systems which have three distinct equilibrium points without symmetry.For some odd Lienard systems,which have damping terms and restore terms in higher degrees of odd numbers,with three distinct equilibrium points without symmetry,We conjecture the sufficient condition of bi-center for these systems.However,we can not prove the necessity of the condition.For some planar differential systems with a parameter,some limit cycles will be bifurcated from the linear center by a small perturbation of the parameter of the systems.It is called the Hopf bifurcation.The Hopf bifurcation is closely related to the study of limit cycles in Hilbert’s sixteenth problem,which is another important topic in qualitative theory of differential systems.For planar polynomial differential systems,there are so many interesting achievements about the number of limit cycles arising from a single weak focus.However,the Hopf bifurcation has not been completely solved even for cubic systems.Many scholars study the Hopf bifurcation perturbed from two weak foci.For this problem,they need to determine not only how many limit cycles that can be perturbed from each weak focus,but also the number of limit cycles that can be perturbed from two weak foci simultaneously.In the second section of Chapter three,we discuss the Hopf bifurcation problem of two weak foci of the cubic Lienard system with three distinct equilibrium points.By some same perturbations,we obtain the styles of the number of the limit cycles arising from these two weak foci.