Phase change and order 2 averaging for one-frequency systems with separatrix crossing

We study the evolution of angular variable (phase) for general (not necessarily Hamiltonian) perturbations of Hamiltonian systems with one degree of freedom near separatrices of the unperturbed system. To this end, we use the averaged system of order 2. We obtain estimates for the accuracy of the order 2 averaged system near separatrices and use these estimates to prove a formula for the phase change when solutions of the perturbed system approach separatrices of the unperturbed system (such formula is known when the perturbation is Hamiltonian). As an application of this formula, we show that two natural de?nitions of probability of capture into different domains after separatrix crossing proposed by V.I. Arnold and D.V. Anosov lead to the same formula for this probability.