Probing the dynamical system and thermal behaviours of the model, $\Lambda_0+3 \beta H^2.$

In this work we study the cosmological evolution of a two component model with non-relativistic dark matter and decaying vacuum of the form $\Lambda = \Lambda_0 + 3 \beta H^2.$ We contrast the model the model with the supernovae data and found that the model parameter $\beta=-0.010$ when the interaction parameter is zero and is $\beta=-0.002$ when $b=0.001.$ The thermal evolution study of the model indicates that it obeys the generalized second of entropy and also satisfies the convexity condition, $\ddot S <0$ so that the model behaves like an ordinary macroscopic system which approaching a stable equilibrium thermal state is the asymptotic condition. The dynamical system analysis reveals that, the model posses a prior decelerated epoch represented by a saddle critical point and the late critical point represents an accelerating epoch which posses a convergence of phase-space trajectories form both the regions of the point hence can be considered as stable.