Classical periodic orbits in extended phase space for spherical harmonic oscillator with spin-orbit coupling

A complete analysis of classical periodic orbits (POs) and their bifurcations was conducted in spherical harmonic oscillator system with spin-orbit coupling. The motion of the spin is explicitly considered using the spin canonical variables derived by semiclassical approximation to the spin coherent state path-integral representation. In addition to the diametric and two circular PO families with frozen spin, solutions that bridge two circular POs are found in which orbital motion is coupled to spin precession. In addition, each bridge encounters a secondary bifurcation on the way from one circular PO to the other and generates a new PO that survives at higher energies while maintaining a constant period. The generic expressions for those POs are obtained explicitly, and all the above peculiar bifurcation scenarios are described fully analytically.