Deviation from Slow-Roll Regime in the EGB Inflationary Models with $r\sim N_e^{-1}$

We consider Einstein-Gauss-Bonnet (EGB) inflationary models using the effective potential approach. We present evolution equations in the slow-roll regime using the effective potential and the tensor-to-scalar ratio. The choice of the effective potential is related to an expression of the spectral index in terms of e-folding number $N_e$. The satisfaction of the slow-roll regime is mostly related to the form of the tensor-to-scalar ratio $r$. The case of $r\sim1/N^2_e$ leads to a generalization of $\alpha$-attractors inflationary parameters to Einstein-Gauss-Bonnet gravity with exponential effective potential. Moreover, the cosmological attractors include models with $r\sim1/N_e$. And we check the satisfaction of the slow-roll regime during inflation for models with $r\sim1/N_e$.