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NONLINEAR STABILITY ANALYSIS OF THE EMDEN–FOWLER EQUATION

Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, United Kingdom

and

Department of Physics and Center for Theoretical and Computational Physics, The University of Hong Kong, Pok Fu Lam Road, Hong Kong, Hong Kong SAR, P. R. China

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https://doi.org/10.1142/S1402925110001100Cited by:44 (Source: Crossref)

Abstract

In this paper, we qualitatively study radial solutions of the semilinear elliptic equation Δu+uⁿ = 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.

Keywords:

AMSC: 34D20, 35B35, 37C75

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