Intrinsic Stability of Periodic Orbits
Abstract
Families of orbits of a conservative, two degreeoffreedom 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, divergencefree 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, divergencefree 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 orbitaveraged 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.
 Publication:

Celestial Mechanics
 Pub Date:
 June 1987
 DOI:
 10.1007/BF01230256
 Bibcode:
 1987CeMec..40..111H
 Keywords:

 Numerical Stability;
 Orbital Mechanics;
 Orbits;
 Degrees Of Freedom;
 Equations Of Motion;
 Potential Flow;
 Riccati Equation;
 Velocity Distribution;
 Physics (General)