Poincare-Dulac method with Chebyshev economization in autonomous mechanical systems simulation problem

We consider an equation of an autonomous dynamical system with one degree of freedom. It contains the linear and cubic forms relative to phase variables. In order to simplify the mathematical model we use the modified asymptotic method of Poincare-Dulac conversion. Using Chebyshev approximations of high-degree polynomials with polynomials of smaller degrees we reduce the residual error. We consider non-oscillatory mechanical systems in the case of absence of internal Poincare resonances that leads to a linear form.