Periodic Orbit Branches Of Several Types Of Differential Systems

This thesis investigates the periodic orbit bifurcations for three classes of differential systems including planar smooth system,high-dimensional piecewise smooth system and planar piecewise smooth system with a curved switching line.For the planar smooth system,the well-known Melnikov function method is used.Furthermore,for the latter two systems,we establish some Melnikov function theories and obtain several interesting results.This thesis is consists of 4 chapters and is organized as follows.We begin the first Chapter by introducing the background,research development and some methods.In the end of this Chapter,our works are proposed.In Chapter 2,a class of near-Hamiltonian system is studied,where the unperturbed system has a double homoclinic loop with second order nilpotent saddle and has three families of periodic orbits around the loop.By investigating the expansions of the first order Melnikov functions near the double homoclinic loop as well as their coefficients,the numbers of limit cycles that may appear around the loop are obtained.To be specific,it is shown that there may exist 11,13,14 or 16 limit cycles under some conditions.Moreover,an example is given to illustrate the theoretical result.The periodic orbit bifurcations of high-dimensional piecewise smooth near-integrable systems are studied in Chapter 3.When the unperturbed system has a family of periodic orbits which are transversal to the switching plane,to investigate the number of periodic orbits of the original system which bifurcated from the family of periodic orbits,we develop a first order Melnikov vector function.In particular,with the aid of this vector function,degenerate Hopf bifurcations and degenerate homoclinic bifurcations can be studied.As an application of the main theoretical result,we investigate a class of 3 dimensional piecewise smooth linear systems and obtain 5 limit cycles.In Chapter 4,the number of limit cycles of planar piecewise smooth near-Hamiltoian systems which has curved switching line is addressed,where the switching line also can be a convex closed curve.When the unperturbed system has a family of periodic orbits,we establish a first order Melnikov function which will play a crucial role in studying the number of limit cycles that bifurcated from the family of periodic orbits.Under the assumption that the boundary of the family of periodic orbits is an elementary center,the formal expansion of the Melnikov function around the center is obtained and a theorem about the Hopf bifurcation is given.As a natural generalization of piecewise smooth near-Hamiltonian system,the first order Melnikov function of planar piecewise smooth near-integral system is studied as well.Finally,as an application,by studying a piecewise smooth linear system whose switching line is a quadratic curve and unperturbed system has a center,we obtain 3 limit cycles near the center.