I have 2d dynamical non-linear system $\dot{x} = f(x)$ with $x\in \mathbb{R}^{2}$ with exactly two stable attractors $x_{1} = ( a\quad b)^{T}$ and $x_{2} = (c\quad d)^{T}$ and one saddle point (One eigenvalue of the linearized system is always positive, and the other one can be positive or negative.) Moreover, I can show that the system never goes to infinity and has no limit cycle, so that it basically reaches one of the attractor.
Is there a way to compute the unstable manifold (or another trick), to know whether the point $(a\quad d)^{T}$ will end-up in $x_{1}$ or in $x_{2}$ ? I can of course do the simulations, but my aim is to find conditions on $f$ such that this is the case.
Many thanks!