How to reach the stable manifold of a saddle node

I need to integrate an ode system which has one equilibrium, which is a saddle point. In particular, I need to start from given initial conditions (that I can adjust) and reach the equilibrium through the stable manifold.

As known, if the initial condition does not exactly lie on the stable manifold, the trajectory diverges away from the equilibrium.

My question: is there any numerical approximation, strategy or odeset option (I'm using ode23) that I can use to reach the equilibrium state by starting from an initial condition which is very close to, but not exactly on, the stable manifold?

Many thanks in advance

Giancarlo