Consider the first order autonomous ode for y(t) a scalar function of one real variable, t:
(i) dy/dt = f(y,c) where c is some real parameter
When 2 equilibrium curves (in the c-y plane) intersect, there are formulas one can use to determine the stability on each curve on either side of the double point (see Iooss "Elementary Stability and Bifurcation Theory", pg 21 or Logan "Applied Mathematics, 2nd edtn, pg 370). In my experience, however, these formulas are very cumbersome to use. It seems much simpler to consider each of the four curves separately, identify the section of each curve between the double point of interest and the next turning or singular point, and then determine the stability at any point on that section of the curve. Since the stability along the equilibrium curve cannot change unless it passes through a turning point or singular point, determining the stability at any point on that section of the curve determines the stability for the entire section. I find that typically one can either find a point at which df/dy can easily be evaluated or use simple asymptotic analysis to determine the sign of df/dy.
I am curious if there is some reason (that perhaps I am missing) why this approach does not seem to be used.
Thank-you, Matt Brenneman