On the Homotopy-First Integral Method for Non-conservative Oscillators

Abstract

This paper presents a ready-to-use formula for determining the number and approximate location of periodic orbits in second-order Lienard systems. As a result of the exact closed-form derived in [16], in which an ordinary differential equation (ODE) must be solved to determine the existence and location of periodic orbits for general non-conservative oscillators, a homotopy functional is defined for Lienard-type systems. This provides a closed-form and ready-to-use polynomial formula with roots as an approximation of the periodic orbit's amplitude. In addition, some examples are analyzed, along with conclusions and future plans.