solution of a dissipative saddle is periodically perturbed, the perturbation either pulls the stable and the unstable manifold of the saddle fix point completely apart, or it creates chaos through homoclinic intersections. In both cases, the separatrix map induced by the solutions of the perturbed equation in the extended phase space is a family of two-dimensional maps. Taking a singular limit, one obtains a family of one-dimensional in the form of (\[f1-s1\]) (with the absolute value sign around $\Phi(x)$ removed). Let $\mu$ be a small parameter representing the magnitude of the perturbation and $\omega$ be the forcing frequency. We have $a \sim \omega \ln
\mu^{-1}$ ([mod]{} $1$),  $L \sim \omega$; and $\Phi$ is the classical Melnikov function (See [@WOk; @WOk11; @WO]).

When we start with [*two*]{} unperturbed homoclinic loops and assume symmetry, then the separatrix maps are a family of annulus maps, the singular limit of which is precisely $f_{a}$ in (\[f1-s1\]) (See [@W]). If the stable and unstable manifolds of the perturbed saddle are pulled completely apart by the forcing function, then $\Phi(x) \neq 0$ for all $x$. In this case we obtain strange attractors, to which the theory of rank one maps developed in [@WY3] apply. If the 