Dynamical system for Tolman–Oppenheimer–Volkoff equation

Robert Stańczy; Dorota Bors

doi:10.3934/dcdsb.2025032

Article Contents

2025, Volume 30, Issue 11: 4482-4497. Doi: 10.3934/dcdsb.2025032

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Dynamical system for Tolman–Oppenheimer–Volkoff equation

Robert Stańczy^1, , and
Dorota Bors^2,

1.
Instytut Matematyczny, Uniwersytet Wrocławski, Poland
2.
Faculty of Mathematics and Computer Science, University of Lodz, Poland

^*Corresponding author: Robert Stańczy
^*Corresponding author: Robert Stańczy

Received: October 2024

Revised: February 2025

Early access: February 19, 2025

Published online: February 19, 2025

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Abstract

The existence of solutions to Tolman–Oppenheimer–Volkoff equation with linear equation of state modeling relativistic cloud of interacting particles is proved for mass parameter below certain threshold. For the intermediate values of mass parameters multiplicity result holds. For the small mass parameter the uniqueness is guaranteed. Moreover, there is a threshold value of the mass, which can not be exceeded. It is achieved by considering the related dynamical system in the rescaled mass–density variables which is governed by the global Lyapunov function with sink. Some preliminary extensions to nonlinear equation of state are discussed with numerical density profiles.

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Mathematics Subject Classification: Primary: 35Q85, 70K05, 85A05; Secondary: 34E15, 37N05.

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Figure 1. Left: the sink for the flow at $ (\frac{1}{16\pi}, \frac{1}{16\pi}) $ governed by the Lyapunov function $ L $; Right: the level sets of the Lyapunov function $ L $; In both pictures the axes refer to the $ x $-horizontal and $ y $-vertical variables

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Figure 2. Left: the heteroclinic orbit joining $ (0, 0) $ with $ (\frac{1}{16\pi}, \frac{1}{16\pi}) $; Right: the heteroclinic orbit along vector field converging to these points; In both pictures the axes refer to the $ x $-horizontal and $ y $-vertical variables

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Figure 3. Left: the heteroclinic orbit joining $ (0, 0) $ with $ (\frac{1}{16\pi}, \frac{1}{16\pi}) $; Right: level set for Lyapunov function; upper triangle with orange exit; In both pictures the axes refer to the $ x $-horizontal and $ y $-vertical variables

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Figure 4. The orbit from $ (0, 0) $ at unstable manifold for $ a = 0.2, 0.1 $ resp. In both pictures the axes refer to the $ x $-horizontal and $ y $-vertical

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Figure 5. The orbit starting from $ (0, 0) $ along unstable manifold in Milne $ (x, y) $ variables with rescaled radius as the argument. Axes: $ x, y $

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Figure 6. The density for $ \rho $ equal: $ 3p $, $ 3p/(1+1/\log p) $, $ 3p+p^{5/7}. $ In all pics horizontal axis refers to radius while the vertical to the density

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