Background
I'm taking a course on differential equations (more theory, less "solve this one").
We're at bifurcations, and I have a gripe with the way it's worded. Both in the text book, and by the teacher's words, bifurcations are described as
drastic changes in behavior given miniscule changes in a parameter
After this we're taught to look for parameter values, to the left and right of which, the equilibrium solutions change type (source, sink, node) or count.
Example
Given the equation $$y' = Ay+y^3$$
we have 3 equilibria when $A<0$, and one when $A>0$. Thus, we have a bifurcation at $A=0$.
Question
Is this it? Is that the extent of the term "bifurcation"? Because if so, I submit that the description above is too vague. "Behavior" seems to me like it could encompass more than just equilibrium points.
No, this is not it. What you are studying is the simplest bifurcations of equilibrium points. There are many other bifurcations. For example, Poincare-Andronov-Hopf bifurcation is about limit cycle being born out of an equilibrium point. There are bifurcations of limit cycles such that an invariant torus appears under a small perturbation of a parameter. These are all examples of local bifurcations, since the changes occur in a small neighborhood of a point or of a limit cycle. There are global bifurcations, the simplest one is appearance of a limit cycle out of a homoclinic trajectory. The list can be easily extended.
So, what is the mathematical definition of a bifurcation? What you are given in your course is not a definition but rather a description. Actually, due to some subtle reasons, it is quite difficult to come up with a universal definition of a bifurcation. The one, which is probably the closest one to being the most general, is that a bifurcation is the appearance of topologically non-equivalent phase portraits under parameter changes. I will leave it to you to google the definition of being topologically equivalent.
There are a lot of higher level books that treat bifurcations from precise mathematical stand point. One of the very few which is rigorous and still geared towards undergraduates is Hale and Kocak Dynamics and Bifurcations. You can find much more details there about your question.