A Tangential Displacement Theory for Locating Perturbed Saddles and Their Manifolds

The stable and unstable manifolds associated with a saddle point in two-dimensional non–area-preserving flows under general time-aperiodic perturbations are examined. An improvement to existing geometric Melnikov theory on the normal displacement of these manifolds is presented. A new theory on the previously neglected tangential displacement is developed. Together, these enable locating the perturbed invariant manifolds to leading order. An easily usable Laplace transform expression for the location of the perturbed time-dependent saddle is also obtained. The theory is illustrated with an application to the Duffing equation.