( ) ( ) ()

If the Mel’nikov’s function (4.1) changes sign, then chaos may occur. In order to apply combined Mel’nikov’s and numerical methods, a perturbation of the Hamiltonian system, where the function sgn(y) occurs, has been approximated by a continuous perturbation with an application of a small parameter. The multivalued relation sgn(y) is approximated by the function ( )y

0 sgn ε defi ned by (3.1), where the regularization parameter 0ε is a “small” positive

real number. The differential equation (inclusion) (2.1) is then approximated by (3.2). In the so-called fi rst improvement of Mel’nikov’s function ( )0τM (see the expression

standing byε ) for 10 << ε in the expression representing a distance between stable and unstable manifolds of the critical saddle point, a transition of the parameter 0ε to zero ( 00 →ε ) can be realized. In order to be sure of neglecting the so-called second improvement of Mel’nikov’s function standing by 2ε [7] (the under integral function includes the differential of the approximated perturbation), the following condition should be satisfi ed

1/ 0 <εε . Then, if the mentioned condition is satisfi ed, only the fi rst improvement of Mel’nikov’s function can be applied to estimate the distance between stable and unstable manifolds of the critical point.