Bifurcations into pathology for Hamiltonian systems

AbstractThis paper presents a geometric analysis of bifurcations leading to chaos for Hamiltonian systems with two degrees of freedom of the form ẋ = y, ẏ = −▽V(x). Two bifurcation parameters are considered. One is the energy level and the other is an angle, Φ, 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 D2V(0) = −I, the 2 × 2 identity matrix. For a fixed energy level it is shown that as Φ decreases through 90 ° 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 R4. 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 manifolds intersect transversely