 Nonlinear crossover delayed by a small unstable-mode coefficient

When the early growing solution is $A{x}^2$ and the nonlinear coefficient depends on $\epsilon y^{\prime }$, its crossover occurs at the displayed scale. If $A$ is order one this is $x=O({\epsilon }^{-1})$; if the initial slope suppresses the leading mode and $A=O(\epsilon )$, it is delayed to $x=O({\epsilon }^{-2})$. The first nonzero perturbative seed must therefore be computed before constructing the global leading approximation; setting the small parameter to zero can erase the mode that ultimately dominates.