Dynamical systems of cosmological models for different possibilities of G and $$\rho _{\Lambda }$$

The present paper deals with the dynamics of spatially flat Friedmann-Lemaitre-Robertson-Walker (FLRW) cosmological model a time varying constant $\Lambda$ where evolves cosmic (t) through Hubble parameter (H). We consider that has reflection symmetry $H \rightarrow -H $ $\Lambda(H)$ expressed in form Taylor series respect to H. Dynamical systems for three different cases based on possibilities gravitational G and vacuum energy density $\rho_{\Lambda}$ have been analysed. In Case I, both are taken be constant. analyse stability system by using notion spectral radius, behavior perturbation along each axis Poincare sphere. II, we dynamical analysis G=constant $\rho_{\Lambda} \neq study concept radius function. III, take $G \neq$ introduce new set variables up corresponding system. find out fixed points from directions: analysing behaviour axis, Center Manifold Theory at infinity sphere respectively. Phase plots presented. deeply scenario obtained late Universe. Our agrees fact Universe is epoch accelerated expansion. EOS $\omega_{eff}$, total $\Omega_{tt}$ also evaluated these values agreement observational [1].