Analytic Approximation of Periodic Ateb Functions via Elementary Functions in Nonlinear Dynamics

We consider the problem of analytic approximation of periodic Ateb functions, which are widely used in nonlinear dynamics. Ateb functions are the result of the following procedure. Initial ordinary differential equation contains only the inertial and non-linear terms. Its integration leads to an implicit solution. To obtain explicit solutions one needs to invert incomplete Beta functions. As a result of this inversion we obtain the special Ateb functions. Their properties are well known, but the use of Ateb functions is difficult on practice. In this regard, the problem arises of approximation of Ateb functions with the help of smooth elementary functions. For this purpose in the present article the asymptotic method is used with a small parameter which is inverted to the exponent of nonlinearity. We also investigated the analytical approximation of Ateb functions' period. Comparison of simulation results, obtained by the approximate expression, with the results of numerical solution of the corresponding Cauchy problem shows their sufficient accuracy for practical purposes, even for the exponent of nonlinearity equal to unity.