A method suitable for studying the exponentially small splitting of separatrices appearing in the generalizations of standard map: \[ x_{1}=x+y_{1},\quad y_{1}=y+\epsilon f(x), \] with \(f\) being a polynomial, trigonometric polynomial, meromorphic or rational function is developed. The method is a combination of analytical and numerical steps, with high-precision computations.
After the introduction, the analytical results on the splitting of separatrices from the generalized standard map are reviewed. Then, full details of numerical methods sketched in C. Simó [in: International conference on differential equations. Proceedings of the conference, Equadiff ’99, Berlin, Germany, August 1–7, 1999. Vol. 2. Singapore: World Scientific. 967–976 (2000; Zbl 0963.65136)] are given. The numerical procedure consists of two main steps: first, the values of the homoclinic invariant are computed; then, the obtained data are used to extract coefficients of an asymptotic expansion. After that, asymptotic formulae for \(f(x)\) being a polynomial of degree \(2\) to \(5\) are described in detail. Finally, singularities of the separatrix solutions of the ODE: \(\ddot{x}_{0}=f(x_{0})\) are studied, both in the case when these solutions can be found explicitly and when this is not possible in terms of elementary functions.