On the problem of total stability for periodic differential systems

Consider a smooth ω-periodic differential system in R×R^n, say S, of ordinary differential equations, and let E be an equilibrium for S. Preliminarily it is shown that the total stability of E is equivalent to the existence of a fundamental family of asymptotically stable neighborhoods of E. Thus a known theorem of Seibert concerning autonomous systems is extended to periodic systems. Let us assume now the existence of a smooth invariant manifold Φ in R×R^n, containing R×{E}, ω-periodic in t, and asymptotically stable near E. By using the above extension of Seibert's theorem and some previous results in a previous paper of us, we prove here that if E is totally stable on Φ (that is with respect to the solutions lying on Φ), then E is unconditionally totally stable.