will be to follow the curve . Extension error estimates Average technique for initial value problems has been treated up to now with an validity error estimates of order . However, there are circumstances where the estimates can be extended for further times, even the case for all times. Below we deal with a system containing an asymptotically stable fixed point. Such situation recapitulates what is illustrated in Figure 1. Theorem (Eckhaus /Sanchez-Palencia ) Consider the initial value problem Suppose exists and contains an asymptotically stable fixed point in the linear approximation. Moreover, is continuously differentiable with respect to in and has a domain of attraction . For any compact and for all with in the general case and in the periodic case. References Dynamical systems