Limit Cycle Bifurcations For Several Classes Of Smooth And Piecewise Smooth Systems

In this paper,the bifurcation problem of limit cycles of several piecewise smooth systems and a class of polynomial systems with small perturbations is studied.It is well known that a powerful tool for studying the number of limit cycles for a Hamiltonian system is the Melnikov function,or Abelian integral.In this paper,the near-integrable system is transformed into an equivalent near-Hamiltonian system.By studying several free coefficients in an expansion of the first-order Melnikov function of the near-Hamiltonian system,the number of limit cycles for several types of systems is estimated.The first chapter is an introduction,which introduces the research background and methods used in this paper.In chapter 2,we discuss the bifurcation of limit cycles for a class of piecewise smooth cubic polynomial systems.By introducing a small parameter ?,using the first two terms of the expansion of the first order Melnikov function on ?,it is obtained that the system can have 13 limit cycles near the center.In Chapter 3,we consider the problem of the number of limit cycles of a class of piecewise smooth cubic polynomial systems under small perturbations of order n.Using the expression of the piecewise smooth Melnikov function near the center at the origin and through complex calculations,it is obtained that the system can have 2n + 1 limit cycles bifurcating from the origin.Further,the Hopf bifurcation of the system is studied when the perturbations does not contain a constant term.We can conclude that the system can produce up to 2n limit cycles under certain conditions,and 2n limit cycles can appear.In Chapter 4,we mainly study the problem of limit cycle bifurcation for a class of cubic polynomial systems with multiple parameters.Because of the analytic property of the first order Melnikov function,it can be expanded with respect to the small parameter ?.By using the second order expansion of Melnikov function on ?,we obtain that the system can have 5 limit cycles bifurcating from the period annulus near the origin if the perturbation is any quadratic polynomial.