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