factors, we can easily obtain all the fixed points of the autonomous system described by Eqs. (\[x’\])-(\[z’\]) and (\[l’\]), and they are shown in Tab. **I**.

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   Fixed-points                     $x_c$                                          $y_c$                        $z_c$   $\lambda_c$                                                     Eigenvalues                                                                              Stability                            
  -------------- ------------------------------------------- ------------------------------------------------- ------- ------------- ----------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------- --
      $P_1$                          $0$                                            $0$                          $0$        $0$                                 $\mathbf{M}^T=(0,-3\gamma,\frac32\gamma,-3+\frac32\gamma)$                                                  U, for all $\gamma$                       

      $P_2$                          $0$                                            $0$                          $0$    $\lambda_*$                             $\mathbf{M}^T=(0,\frac32\gamma,-3\gamma,-3+\frac32\gamma)$                                                  U, for all $\gamma$                       

                                                                                                                                                                                                                                                                       S, for $\gamma=1,f_1(0)\geq0$                  

      $P_3$                          $0$                                            $1$                           0         $0$                                              $\mathbf{M}^T=(-3,-3\gamma,0,0)$                                                    U, for $\gamma=\frac43$, if $f_1(0)\neq 0$           

                                                                                                                                                                                                                                                                   S, for $\gamma=\frac43$, if $f_1(0)=0$             

      $P_4$                          $1$                                            $0$                          $0$        $0$                                              $\mathbf{M}^T=(0,-6,0,6-3\gamma)$                                                               U,for all $\gamma$                       

      $P_5$                         $-1$                                            $0$                          $0$        $0$                                              $\mathbf{M}^T=(0,-6,0,6-3\gamma)$                                                               U,for all $\gamma$                       

      $P_6$                          $0$                                            $0$                          $0$    $\lambda_a$                             $\mathbf{M}^T=(0,\frac32\gamma,-3\gamma,-3+\frac32\gamma)$                                                   U,for all $\gamma$                       

      $P_7$                          $1$                                            $0$                          $0$    $\lambda_a$                                                   $\mathbf{M}^T=                                                                         U,for all $\gamma$                       
                                                                                                                                                         \left(-6,6-3\gamma,\frac12\sqrt{6}\lambda_a+3,\sqrt{6}\lambda_a A\right)$                                                                                    

      $P_8$                         $-1$                                            $0$                          $0$    $\lambda_a$                                                   $\mathbf{M}^T=                                                                         U,for all $\gamma$                       
                                                                                                                                                        \left(-6,6-3\gamma,-\frac12\sqrt{6}\lambda_a+3,-\sqrt{6}\lambda_aA\right)$                                                                                    

      $P_9$            $-\frac{\sqrt{6}}{6}\lambda_a$                $\sqrt{1-\frac{\lambda^2_a}{6}}$             0     $\lambda_a$   $\mathbf{M}^T=\left(-\lambda_a^2,-3+\frac12\lambda_a^2,\lambda_a^2-3\gamma,-\lambda^3_a-f_1(\lambda_a) \right)$    S, for $f_1(\lambda_a)>\lambda_a$, and $\lambda_a<3\gamma$   

                                                                                                                                                                                                                                                        U, for $f_1(\lambda_a)<\lambda_a$ and/or $\lambda_a>3\gamma$  

     $P_{10}$     $-\sqrt{\frac32}\frac{\gamma}{\lambda_a}$   $\sqrt{\frac{3}{2\lambda_a^2}\gamma(2-\gamma)}$     0     $\lambda_a$                                                See the Eq. (\[p10\])                                                                    U, for all $\gamma$                       
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  : The stability analysis of an autonomous system in LQC. The system is described by a self-interacting scalar field $\phi$ with positive potential $V$ coupled with a barotropic fluid $\rho_\gamma$. Explanation of the symbols used in this table: $P_{i}$ 