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Non-minimally coupled quintessence DE model with a cubic galileon term --A Dynamical System Analysis

We consider a scalar field which is generallly non-minimally coupled to gravity and has a characteristic cubic Galilean-like term in the kinetic part of the action, in presence of a generic self-interaction as a candidate Dark Energy model. The system is dynamically analyzed and novel fixed points with perturbative stability are demonstrated. Evolution of the system is numerically studied near a novel fixed point which owes its existance to the Galileon character of the model. It turns out that demanding the stability of this novel fixed points puts strong restriction on the allowed non-minimal coupling and the choice of the self-interaction. The evolutions of the system is charted out on a $r-s$ diagram. The evolution of the equation of state parameter is studied which shows that our model predicts accelerated universe throughout and the phantom limit is only approached closely but never crossed. Our result thus extends the findings of of \cite{Cubic_Galileon_NMC} for more general NMC than linear and quadratic couplings.