In many strange attractors (for example the Lorentz system, given appropriate parameter values), the point that is described by the system's equations of motion seems to approach a manifold with a dimensionality lower than that of the point as $t\to\infty$. (For example, in the Lorentz system, the point has three dimensions as it is described by $x$, $y$ and $z$, but the manifold to which it eventually ends up arbitrarily close only seems to have two dimensions.) If the point is in this manifold, the system can be considered to be in a kind of equilibrium, as the point will never stray from this manifold as time progresses (similarly to how the entropy of a physical system in equilibrium will remain constant).
My question is, is there some algorithm that can be used to quickly generate an equilibrium point of the system?
I don't consider a stationary point an equilibrium point as it has a very low entropy (nor would I consider any other point with significantly lower entropy than the entropy of the manifold an equilibrium point). Obviously, the stationary point is also part of the manifold, but if our strategy is to just find a stationary point, we limit ourselves to a subset of the manifold, and since this subset contains much fewer points than what the entire manifold does, the entropy of the solutions that we find using this strategy becomes much smaller.
One way to generate an equilibrium point is to simply simulate the trajectory of the point, using a numerical ODE solver, sufficiently far into the future. (In fact, a more rigorous way to define the set of equilibrium points may be as the limit set $\lim_{T\to\infty} \{\vec{u}(\vec{u}_0,t)\,|\,t\geq T,\, \vec{u}_0\in U \}$, where $U$ is the set of all starting points that are well-behaved according to some appropriate criterion, or maybe it would even be enough if $U$ just was a set containing a single well-behaved starting point.) However, this feels very inefficient, and I'm looking for a much quicker way to get to equilibrium.
I then asked myself: Can we device an equation that describes the manifold? That is, an equation that is true if and only if a given point lies within the manifold? If so, we could perhaps generate a point at random, outside of the manifold, and then either use use gradient decent or the nonlinear conjugate gradient method in order to force the value $(\text{LHS} - \text{RHS})^2$ to become zero by pushing the point more directly towards the manifold, and in that case this method could probably be used to reach an equilibrium point much faster. It would therefore also be nice to know whether we can describe the manifold by using an equation (rather than by using the much less practical limit set).