Existence of periodic orbits in nonlinear oscillators of Emden-Fowler form

TL;DR

This paper analyzes the existence of periodic solutions in Emden-Fowler type nonlinear oscillators by phase-space analysis, invariant transformations, and integrability conditions, providing new criteria for periodic orbits.

Contribution

It introduces a method to identify and construct periodic solutions in EF equations using invariant transformations and integrability conditions, expanding understanding of nonlinear oscillator behavior.

Findings

Periodic solutions exist under specific integrability conditions.

Invariant transformations can convert non-integrable EF equations into integrable forms.

The analysis applies to Emden-Fowler, Ermakov, and related nonlinear equations.

Abstract

The nonlinear pseudo-oscillator recently tackled by Gadella and Lara is mapped to an Emden-Fowler (EF) equation that is written as an autonomous two-dimensional ODE system for which we provide the phase-space analysis and the parametric solution. Through an invariant transformation we find periodic solutions to a certain class of EF equations that pass an integrability condition. We show that this condition is necessary to have periodic solutions and via the ODE analysis we also find the sufficient condition for periodic orbits. EF equations that do not pass integrability conditions can be made integrable via an invariant transformation which also allows us to construct periodic solutions to them. Two other nonlinear equations, a zero-frequency Ermakov equation and a positive power Emden-Fowler equation are discussed in the same context