Intrinsic stability of periodic orbits

Families of orbits of a conservative, two degree-of-freedom system are represented by an unsteady velocity field with componentsu(x, y, t) andv(x, y, t). Intrinsic stability properties depend on velocity field divergence and curl, whose dynamical evolution is determined by a matrix Riccati equation. Near equilibrium, divergence-free or irrotational fields are dynamically compatible with the conservative force field. It is shown that a necessary condition for stable periodic orbits is satisfied when the orbitaveraged divergence is zero, which results in bounded normal variations. A sufficient condition for stability is derived from the requirement that tangential variations do not exhibit secular growth.In a steady, divergence-free field, velocity component functionsu(x, y) andv(x, y) may be continuedanalytically from any initial condition, except when velocity is parallel to U or at equilibria. In an unsteady field, the orbit-averaged divergence is zero when the vorticity function is periodic. When such a field exists, initial conditions for stable periodic orbits (i.e., characteristic loci) may be determinedanalytically.