satisfies your criteria but not the form g(y) +a. I'm not sure one can exist.
Is it possible to find a function g(y) (either continuous or discontinuous) such that the one-parameter family of differential equations
satisfies both the following statements?
for all a<= -1, the differential equation has exactly one equilibrium point which is a sink.
for all a>= 1, the differential equation has exactly three equalibria , two sources and one sink.
If so give a rough sketch, if not, why not?
I've sketched a ton of graphs now and can't get one to satisfy, also I have no idea how to explain then if it don't exist, why that is.
Thanks in advance.
Other than guessing is there a way for me to determine possible functions?
Oh I didn't even notice that. My answer was for both equalities -1. Basically I've only ever seen a single eq. splitting to 3 by having a factor of y through the whole thing as then you get y=0 for all a. If you solve the eqn g(y)=-a you need a function that has one solution for a<-1 and 3 for a>1. this isn't possible if you want g to be continuously differentiable (or even continuous only I think).