We analyze the dynamical system defined by a universe filled with a barotropic fluid plus a scalar field with modified kinetic term of the form L = F(X) − V (phi). After a suitable choice of variables that allows us to study the phase space of the system we obtain the critical points and their stability. We present a particular model that possesses a large set of initial conditions consistent with a successful cosmological model in which an inflationary phase is possible, followed by a matter era to end with dark energy domination. In order to understand the success of the model, we then study the general features that unified dark matter (UDM) models should comply and analyze some particular models and find their constrictions.