I am looking for an efficient, non-exponential time algorithm to calculate the boundary of a basin of attraction for a stable fixed point in a high dimensional nonlinear dynamical system. The naive method (simulating the system, finding points "close" to other points in a different basin) appears to require an exponentially increasing number of points to maintain a similar density of points (since the basin boundary is expected to be an $N-1$ dimensional manifold), and is thus not really feasible.
Alternatively, if anyone knows of something saying this isn't possible, that would be useful too.
Thanks!