Periodic Solutions of a Perturbed Autonomous System

for small s. We shall obtain results pertaining to existence, uniqueness, and stability of solutions of (1.1) in the form (1.3). Some of the results are illustrated with van der Pol's equation. Levinson [4] has discussed the case of (1.1) when x0(t) is strongly stable (cf., Section 4). He also makes no assumption about the ratio of T to Lo. His results show that for small 6 all solutions of (1.1), regarded as trajectories in (n + 1)-dimensional space, tend to a hyper-surface which lies near the hyper-cylinder having x = xo(t) as base, provided that at some finite time they are near enough to the hyper-surface. This result shows that even though some of the periodic solutions we find in the present paper are unstable, their instability is of a very weak type when x0(t) is strongly stable. The author [5] has made a study similar to the present paper of the second-order scalar equation