The Stablity And Monotonicity Of Periodic Orbits Of The Planar Hamlitonian System

Differential equations are important ways when applying mathematic theory to reality. Twenty centuries before, many of the problems of differential equations originated from Geometry, Mechanics and Physics. Now they emerge from almost every field of science of nature and technology of engineering. So the problems from theory of differential equations are various.Hamiltonian system is a important system in theory of differential equations and in nonlinear science research. An integrated Hamiltonian theory was constituted in Classical Mechanics and applied to studying the form of Lagrange equation in theory of electromagnetic, the Schrodinger equation and the Liouville theorem in statistical Physics. The studies of Hamiltonian system simplified from the reality are abroad in Mathematics. Consider the subharmonic bifurcation, a hypothesis of the monotonicity of periodic orbits is always needed. Thus the study of this monotonicity develops rapid in recent 20 years. In the first chapter, some knowledge about Hamiltonian system, the theory of stability and monotonicity and isochronism of periodic orbits are given.In Chapter 2, an autonomous perturbation of a class of planar Hamiltonian systems with homoclinic orbits is discussed. Under appropriate conditions the existence and the continuous dependence of its homoclinic orbits and periodic orbits on the perturbing small function are obtained.When discussing the bifurcation of subharmonics orbits, many papers need a samilar assumption of the monotonicity of periodic orbits near the homoclinic orbit. In Chapter 3, a certain polynomial Hamilton system with 3 degree is given. By elliptic integral formulas, the monotonicity of the periodic orbits of this system is verified.