Equivalent linearization finds nonzero frequency corrections beyond first order

TL;DR

This paper demonstrates that equivalent linearization can be effectively used to compute higher-order frequency corrections in weakly nonlinear oscillators, surpassing the limitations of first-order approximations.

Contribution

It introduces a general method for calculating second-order and higher frequency corrections using equivalent linearization, applicable to various types of oscillators.

Findings

Successfully applied to anharmonic oscillators and van der Pol oscillator

Shows that higher-order corrections can be nonzero even when first-order corrections vanish

Provides a versatile approach for analyzing nonlinear oscillatory systems

Abstract

We show that the equivalent linearization technique, when used properly, enables us to calculate frequency corrections of weakly nonlinear oscillators beyond the first order in nonlinearity. We illustrate the method by applying it to the conservative anharmonic oscillators and the nonconservative van der Pol oscillator that are respectively paradigmatic systems for modeling center-type oscillatory states and limit cycle type oscillatory states. The choice of these systems is also prompted by the fact that first order frequency corrections may vanish for both these types of oscillators, thereby rendering the calculation of the higher order corrections rather important. The method presented herein is very general in nature and, hence, in principle applicable to any arbitrary periodic oscillator.