Bifurcations into Pathology for Hamiltonian Systems

摘要

This paper presents a geometric analysis of bifurcations leading to chaos for Hamiltonian systems with two degrees of freedom of the form x-dot = y, y-dot = -gradient V(x). Two bifurcation parameters are considered. One is the energy level and the other is an angle, Psi, between two homoclinic orbits. Though global non-linearities are necessary, the results are obtained by local analysis of the flow near the origin where it is assumed that (D-sq)V(0) = I, the 2 x 2 identity matrix. For a fixed energy level it is shown that as Psi decreases through 90 deg the two homoclinic orbits bifurcate into two homoclinic orbits, a periodic orbit, and connecting orbits. These orbits can then be used to define a compact region in R supercript 4. Now treating the energy as a parameter value the trajectory of orbits passing through this compact region can be described using symbolic dynamics. In this case it is shown that a single periodic orbit bifurcates into three periodic orbits whose stable and unstable manifold intersect transversely.