Nonlinear stability analysis of the Emden-Fowler equation

TL;DR

This paper investigates the stability of solutions to the Emden-Fowler equation, a key model in astrophysics, using various stability analysis methods to identify parameter ranges of stability.

Contribution

It introduces a comprehensive stability analysis framework combining linear, Jacobi, and Lyapunov methods for the Emden-Fowler equation, revealing parameter-dependent stability conditions.

Findings

Identifies parameter ranges where solutions are stable using multiple methods.

Demonstrates differences in stability results depending on the analysis technique.

Provides a unified approach to stability analysis of nonlinear differential equations.

Abstract

In this paper we qualitatively study radial solutions of the semilinear elliptic equation $\Delta u + u^n = 0$ with $u(0)=1$ and $u'(0)=0$ on the positive real line, called the Emden-Fowler or Lane-Emden equation. This equation is of great importance in Newtonian astrophysics and the constant $n$ is called the polytropic index. By introducing a set of new variables, the Emden-Fowler equation can be written as an autonomous system of two ordinary differential equations which can be analyzed using linear and nonlinear stability analysis. We perform the study of stability by using linear stability analysis, the Jacobi stability analysis (Kosambi-Cartan-Chern theory) and the Lyapunov function method. Depending on the values of $n$ these different methods yield different results. We identify a parameter range for $n$ where all three methods imply stability.