The phenomenon of separatrix splitting and complicated homoclinic structures constitute a symptom of chaos, as pointed out by Poincaré more than a century ago. Taking a time-discrete dynamical system with a double-well potential as an example, this interesting feature is demonstrated analytically on the basis of asymptotic expansions beyond all orders. Violent undulations of the unstable manifold in the extreme vicinity of hyperbolic fixed points are recovered excellently. Comparison with results for standard and Henon maps is also made.