identify an unstable manifold of initial data which leads to a globally decaying solution to - . For the present discussion, we focus on a model ODE problem. We consider $$\begin{aligned}
 \label{ODE:model}
({\ensuremath{\partial}}_s - \frac 12) u_{\alpha} = g + \eps f(u_{\alpha}), \qquad u_\alpha(0) = \alpha\,. \end{aligned}$$

We assume for now that the forcing, $g$, has sufficiently strong decay and the nonlinearity, $f$, is an explicit quadratic nonlinearity via $$\begin{aligned}
 \label{as:setup}
|g| \lesssim e^{- \gamma s}, \qquad f(u) = u^2, \qquad \gamma > 0\,. \end{aligned}$$

For general data, $\alpha$, writing the solution to via the Duhamel formula yields $$\begin{aligned}
 \label{aloha:1}
u_\alpha(s) = e^{\frac s 2} \alpha + e^{\frac s 2} \int_0^s e^{- \frac{s'}{2}} g(s') {\,\mathrm{d}}s' + \eps e^{\frac s 2} \int_0^s e^{- \frac{s'}{2}} u_\alpha(s')^2 {\,\mathrm{d}}s'\,. \end{aligned}$$ From , it is that even under the assumption of $g$ decaying exponentially one cannot expect the solution $u_\alpha$ to decay to zero as $s \rightarrow \infty$ for *generic data*, $\alpha$. Thus, to obtain decaying solutions to , one needs to find an unstable manifold of data. In the case of this ODE, this amounts to finding a *particular value* of $\alpha$ which ensures a globally decaying solution.

To illustrate how to find this 