Periodic orbits and their bifurcations in a 3-D system

Abstract

We study some simple periodic orbits and their bifurcations in the Hamiltonian$H = \tfrac{1}{2}(\dot x^2 + \dot y^2 + \dot z^2 + Ax^2 + By^2 + Cz^2 ) - \varepsilon xz^2 - \eta yz^2$. We give the forms of the orbits, the characteristics of the main families, and some existence diagrams and stability diagrams. The existence diagram of the family 1a contains regions that are stable (S), simply unstable (U), doubly unstable (DU) and complex unstable (Δ). In the regionsS andU there are lines of equal rotation numberm/n. Along these lines we have bifurcations of families of periodic orbits of multiplicityn. When these lines reach the boundary of the complex unstable region, they are tangent to it. Inside the region Δ there are linesm/n, along which the orbits 1a, describedn-times, are doubly unstable; however, along these lines there are no bifurcations ofn-ple periodic orbits. The families bifurcating from 1a exist only in certain regions of the parameter space (ε, η). The limiting lines of these regions join at particular points representing collisions of bifurcations. These collisions of bifurcations produce a nonuniqueness of the various families of periodic orbits. The complicated structure of the various bifurcations can be understood by constructing appropriate stability diagrams.