On the small--amplitude approximation to the differential equation $\ddot{x}+(1+\dot{x}^{2})x=0$

TL;DR

This paper analyzes the small-amplitude approximation for a nonlinear oscillator, determining its convergence radius and showing the inverted perturbation series converges smoothly from below.

Contribution

It provides the first detailed analysis of the convergence properties of the small-amplitude series for this specific nonlinear oscillator.

Findings

Radius of convergence of the approximation is explicitly obtained.

Inverted perturbation series converges smoothly from below.

Enhanced understanding of nonlinear oscillator perturbation series.

Abstract

We obtain the radius of convergence of the small--amplitude approximation to the period of the nonlinear oscillator $\ddot{x}+(1+\dot{x}^{2})x=0$ with the initial conditions $x(0)=A$ and $\dot{x}(0)=0$ and show that the inverted perturbation series appears to converge smoothly from below.