  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.
