Then, treating as a small parameter, we rescale time by letting t = . We may then introduce the following rescaled parameters: ␣ = ␣ 0 / and ␥ = ␥ 0 / , where ␣ 0 and ␥ 0 are O͑1͒. The inclusion of the parameter as a new state variable means that the terms in our rescaled system which contain are now nonlinear terms. Furthermore, the system is augmented with the auxiliary equation d / d = 0. The addition of this auxiliary equation contributes an extra simple zero eigenvalue to the system and adds one new center direction that has trivial dynamics. The shifted and rescaled augmented system of equations is