THE STABILITY OF PERIODIC REVERSIBLE-SYSTEMS

Some results established in [1-4] for autonomous reversible systems ar e extended to periodic systems that are reversible under the substitut ion t --> -t, x --> M(-t)x, M2(t) = E (where E is the identity matrix) , both in the non-resonant case and in the case of internal resonance. A complete solution of the problem of stability in the first approxim ation is derived for single-frequency resonances, which have no analog ue in autonomous systems. It turns out that a system with one degree o f freedom is either unstable or stable to any finite order. It is show n that the known conditions 15] do in fact guarantee formal stability in the rolling of a heavy, homogeneous, almost symmetric ellipsoid in the principal plane.