Dynamics of the Isotropic Star Differential System from the Mathematical and Physical Point of Views

• Published in: AppliedMath 2024-03, Vol.4 (1), p.70-78
• Main Authors: Artés, Joan-Carles, Llibre, Jaume, Vulpe, Nicolae
• Format: Article
• Language: English
• Subjects: isotropic star; phase portrait; Poincaré disc; polynomial differential equation
• ISSN: 2673-9909; 2673-9909
• DOI: 10.3390/appliedmath4010004

The following differential quadratic polynomial differential system  dxdt=y−x, dydt=2y−yγ−12−γy−5γ−4γ−1x, when the parameter γ∈(1,2] models the structure equations of an isotropic star having a linear barotropic equation of state, being x=m(r)/r where m(r)≥0 is the mass inside the sphere of radius r of the star, y=4πr2ρ where ρ is the density of the star, and t=ln(r/R) where R is the radius of the star. First, we classify all the topologically non-equivalent phase portraits in the Poincaré disc of these quadratic polynomial differential systems for all values of the parameter γ∈R∖{1}. Second, using the information of the different phase portraits obtained we classify the possible limit values of m(r)/r and 4πr2ρ of an isotropic star when r decreases.