Bifurcation of periodic orbits for non-positive definite hamiltonian systems

Abstract

We consider the bifurcations of periodic solutions of a family of non–positive definite Hamiltonian systems of n degrees of freedom near the origin as the family passes through a semisimple resonance, We begin with a smooth Hamiltonian H with a general semisimple quadratic part H₂ and then construct a normal form of H with respect to Hg up to fourth order terms and make a versal deformation. We apply the Liapunov–Schmidt reduction in the presence of symmetry and further reduce the resulting bifurcation equation to a gradient system. Thus, the study of periodic solutions of the original system is reduced to finding critical points of a real-valued function. As an application, we consider a system with two degrees of freedom in 1:−1 semisimple resonance by using suitable choices of the parameters to study the bifurcation as the eigenvalues split along the imaginary axis or across it and we obtain complete bifurcation patterns of periodic orbits on each energy level

Keywords:

• Bifurcation
• periodic orbits
• Hamiltonian systems
• non-positive definite
• normal forms
• Liapunov center theorem
• Liapunov-Schmidt method

AMS(MOS)::

• 34C25
• 70H05

^∗Partially supported by DARPA and NSF grant DMS 8401719

^∗Partially supported by DARPA and NSF grant DMS 8401719

Notes

^∗Partially supported by DARPA and NSF grant DMS 8401719