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

We classify all the topologically non-equivalent phase portraits of the quadratic polynomial differential system dx/dt = (1-2x)(y-x), dy/dt =y (2-g y-(5g-4)x/(g-1)), in the Poincaré disc for all the values of the parameter g in R\{1}. The differential system dx/dt = y-x, dy/dt =y (2-g y-(5g-4)x/(g-1))/(1-2x), when the parameter g in (1,2] models the structure equations of a static star in general relativity in the case of the existence of a homologous family of solutions, being x = m(r)/r where m(r)>= 0 is the mass inside the sphere of radius r of the star, y = 4pi r^2 rho where rho is the density of the star, and t = ln (r/R) where R is the radius of the star. We classify the possible values of m(r)/r and 4pi r^2 rho when r-->0.