I just have a question regarding equilibrium points and bifurcation diagrams. This is the question that we are being asked;
For the following models, which contain a parameter $h$, find the equilibria in terms of $h$ and determine their stability. Construct a bifurcation diagram showing how the equilibria depend upon $h$. and label the branches of the curves in the diagram as stable or unstable. $$1.\space\space\space\space u'=hu-u^2$$ $$2.\space\space\space\space u'=(1-u)(u^2-h)$$
I have determined the equilibrium solutions in terms of $h$. For 1. $h=u$ and for 2. $h=u^2$
When determining their stability do we assume that $h$ is just some constant and determine the second derivative with respect to $h$ and continue by substituting our equilibrium value in? Or is there something else that we need to do.
I have done a bit of searching for bifurcation diagrams but I have not found any way that I can plot it with the materials available. By any chance is there some program that I could use to display this on a diagram?
Any help would be greatly appreciated! Thanks in advance.