A Generalization of Lindstedt–Poincaré Perturbation Method to Strongly Mixed-Parity Nonlinear Oscillators

Abstract

In this paper, generalization for the Lindstedt–Poincare perturbation method for nonlinear oscillators to a class of strongly mixed-parity oscillating system is established. In this extended and enhanced approach, two new odd nonlinear oscillators are introduced in terms of the mixed-parity oscillator. By combining the analytical approximate solutions corresponding to the two new systems, the accurate approximate solutions of the original mixed-parity oscillator are obtained. Comparing with the existing methods such as the perturbation method, the new solution methodology for the singular nonlinear system introduced is not only simple, but the combinatory solution is straight forward and it yields very accurate and physically insightful solutions. Using two typical examples, we demonstrated that this new proposed approach is capable of establishing highly accurate approximate analytical frequency and periodic solutions for small as well as large amplitude of oscillation. The new analytical methodology established will potentially shed new insights to the physical interpretations of strongly nonlinear oscillators including optoelectronic oscillators, pendulums and spring-masses.