How bistable is my system?

Question

Description: given ODE: $\dot{x} = a + bx +cx^2 +dx^3$,
I have mutliple combinations of the coefficients $a,b,c,d$ that I want to understand whether they make a bistable or not system. For this purpose, I started with the discriminant $\Delta$ of the cubic equation for $\dot{x}=0$, which:

1. for $\Delta > 0$, 3 real solutions/fixed points (I guess 2 stable and 1 unstable)
2. for $\Delta = 0$, 1 and 1 double real solutions/fixed points (stability ?)
3. for $\Delta<0$, 1 real solution/fixed point and 2 imaginary.

So, using $\Delta > 0$ I am able to find numerically which combinations of coefficients give me 3 real solutions and a clearly bistable system. But how strongly bistable are my systems?

My current approach: I made the leap to hypothesize that the more positive my $\Delta$ is, the more bistable my system is. Basically I thought that because it seems that $\Delta$ is actually a metric of separating the solutions, since for $\Delta=0$ the double solution is just two collapsed solutions to the same point. So, it actually feels like $\Delta$ is like a control parameter of the system, which by varying it you cross bifurcation points and generate solutions. However, probably sth like that is not proved and it is like my approach makes some sort of sense but it is illegitimate. So, any comments on that would be highly appreciated. Energy landscapes ($-\int\dot{x}dx$) for different combinations of $a,b,c,d$ and a range of positive $\Delta$

Another approach: More legitimate I thought that a proper stability analysis would be. So that would be to find numerical solutions of $\dot{x}=0$, and do linearized stability analysis (get the Jacobian over the fixed points) to see the nature of the fixed points. But, my next question is, what should I look for to judge how strong the bistable system is? Is it how positive the unstable fixed point is (greater instability of the fixed point, more bistable)? Or is it a combination with how negative the stable fixed points are?

Other approaches: At the end, I am not sure whether some sort of statistical significance could be made to determine how strong bistability exists (and stats is my weakest quality).