Melnikov's vector - a ``Measure of chaos'' (Q1201387)

An asymptotic perturbation theory, based on the Melnikov's function is developed for nonlinear dynamical systems with the left-hand sides in the form of integrable zeroth-order terms plus a small perturbation (which may be both Hamiltonian and non-Hamiltonian). It is assumed that the integrable (zeroth-order) dynamical system has an exact homoclinic trajectory, or a closed cycle composed of several heteroclinic trajectories. The perturbation theory developed makes it possible to detect the presence of the Smale horseshoe in the phase space of the perturbed system, i.e., to predict appearance of the dynamical chaos in a vicinity of the homoclinic trajectory, generated by the small perturbation.