Set of unstable initial conditions

Suppose I have some equations of motion for the variables $x(t),y(t)$
begin{align}
0&=ddot x + b(t) dot x + V_x(x,y),,
0&=ddot y + b(t) dot y + V_y(x,y),,
end{align}
where the potential $V$ has a saddle point at, say $x = y = 0$. Is there an efficient formalism to compute set of initial conditions $x(0),y(0),dot x(0),dot y(0)$ such that $x(infty) = y(infty) = dot x(infty) = dot y(infty) = 0$? Or, is there a physical way to pose this problem that makes the solution intuitive? Numerically this problem is not hard, but I think there should be a nice way to phrase the problem that makes the solution better motivated than "pure numerics."