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In this paper, we consider a normal branch of the DGP cosmological model witha quintessence scalar field on the brane as the dark energy component. Usingthe dynamical system approach, we study the stability properties of the model.We find that {\lambda}, as one of our new dimensionless variables which isdefined in terms of the quintessence potential, has a crucial role in thehistory of the universe. We divide our discussion into two parts: a constant{\lambda} and a varying {\lambda}. In the case of a constant {\lambda}, wecalculate all the critical points of the model even those at infinity and thenassume all of them as instantaneous critical points in the varying {\lambda}situation which is the main part of this paper. We find that the effect of theextra dimension in such a model is independent of the value of {\lambda}. Then,we consider a Gaussian potential for which {\lambda} is not constant but variesfrom zero to infinity. We discuss the evolution of the dynamical variables ofthe model and conclude that their asymptotic behaviors follow the trajectoriesof the moving critical points. Also, we find two different possible fates forthe universe. In one of them, it could experience an accelerated expansion, butthen enters a decelerating phase and finally reaches a stable matter-dominatedsolution. In the other scenario, the universe could approach thematter-dominated critical point without experiencing any accelerated expansion.We argue that the first scenario is more compatible with observations.

The Stability Analysis of Brane Induced Gravity with Quintessence Field on the Brane with a Gaussian Potential