2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/15697We describe and characterize rigorously the chaotic behavior of the sine-Gordon equation. The existence of invariant manifolds and the persistence of homoclinic orbits for a perturbed sine--Gordon equation are established. We apply a geometric method based on Mel'nikov's analysis to derive conditions for the transversal intersection of invariant manifolds of a hyperbolic point of the perturbed Poincare map.25 pages, LaTeX, no figuresChaotic Dynamicstext