Several Problems About Homoclinic Bifurcation In The Quadratic Systems

So far it is still difficult to determine the homoclinic (or heteroclinic)bifurcation in the quadratic systems and there are many reminded questiones about the algebraic solvable part of this problem. This thesis devotes to study the homoclinics of quadratic systems and their revelent problems.The content is divided to four parts.In the first part, the generic quadratic systems with a hyperbolic saddle are discussed. By transforming the systems to a certain normal form, we obtain some necessary conditions for the systems possessing the homoclinic cycle.In the second part, we discuss two kinds of integrable systems with a hyperbolic saddle. One is symmetric integrable system,the other is hamiltonian integrable system.We analysis in detail the conditions of the systems possessing homoclinic cycles,then give all of the complete bifurcation curves in the parameter plane and the global phase graphs of the systems.In the third part,we study the question of parameter unfolding of a simple hamiltonian system.By choosing the proper parameters, we can get a four-dimension parameter domain which ensures the unfolded system have the limit cycle .The boundaries of the domains corresponding to poincdre, homoclinic, het-eroclinic, hopf or saddle-node bifurcation of the systems.Finally,we present a kind of quintic curve whose non-isolated component can constitute the homoclinic cycle of the quadratic systems.