 Period near a separatrix

A bound orbit approaching a separatrix can spend an increasingly long time near an unstable equilibrium. Suppose a one-dimensional potential energy has a nondegenerate maximum at $x_s$, so locallyFor energy just below $U_s$, the nearby turning point approaches $x_s$. The travel-time integral contains a contribution proportional to $\int dy/\sqrt{{\lambda }^2{y}^2-2(U_s-E)/m}$, whose growth is logarithmic as $E\to U_s$.

A simple pendulum illustrates this slowing: a release increasingly close to the upright position produces a long pause near the top of each swing. The full period of an oscillation diverges, although motion away from the unstable point remains fast. The logarithm relies on a nonzero negative curvature at the maximum; a flatter maximum can give a different divergence.