The Poincaré-Mel'nikov geometric analysis of the transversal splitting of manifolds of slowly perturbed nonlinear dynamical systems. I (Q1344995)

Given an autonomous differential system with a heteroclinic orbit between two hyperbolic singular points, a \((2\pi)\)-periodic perturbation with parameter \(\varepsilon\) is studied. Suitable conditions on the Poincaré-Mel'nikov vector function at \(\varepsilon = 0\) guarantee transversal intersection of the local stable manifold of one and the local unstable manifold of the other singular point for small non-zero \(|\varepsilon |\). For a Hamiltonian system, this may lead to a case of destruction of the heteroclinic structure for \(\varepsilon \to 0\).