The convergence of the slaving principle in a simplified model

We prove the convergence of the slaving principle in a model consisting of two nonlinear equations with two variables,s andu, which represent in the linear regime the stable mode and unstable mode, respectively.We show explicitly how the stable modes becomes increasingly dependent on the unstable modeu and approaches a definite power series ofu regardless of the initial condition fors. This power series is called slaving function and is shown to be absolutely and uniformly convergent on a closed disc, which contains the point describing the asymptotic behavior of the system. For some finite time, we show that the approximation involved in the substitution of the slaving function for the original stable modes decreases exponentially with time.