Slow-roll trajectories in Chromo-Natural and Gauge-flation Models, an exhaustive analysis

We present an exhaustive analysis on the classical solutions of the Chromo-natural model. We showed that starting from an arbitrary axion field value ${\chi }_0\in (0,f\pi )$, it is possible to have successful slow-roll inflation and working out the analytical solution of the number of e-folding, we determine the allowed region of the parameters corresponding to each ${\chi }_0$ value. Having the allowed parameter region, then we study the behavior of the solutions respect to the initial value of the axion field.