# In what sense do bifurcations concern change in quality?

I've heard such vague statements several times and also read:

Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family.

(From Bifurcation theory in Wikipedia.)

In what sense do bifurcations concern change in quality?

• Your question seems to have it backward. You ask about "bifurcation creating" whereas the statement in your question (which corresponds, more or less, to my understanding) defines it as the study of changes that already exist. – Brick Jun 28 at 16:48
• You might be right about that. I'm going to change it! – SomeGuy Jun 28 at 16:52

## 1 Answer

A qualitative change is a change in the nature of the behavior, in a defining feature of it. It's therefore deeper, more important than a quantitative change, by which it's meant that specific numerical values that characterize the system might change, but they still define roughly the same type of behavior.

Bifurcations describe qualitative, deeper changes, such as in the number and/or type of stable solutions of the system. For example, if a 1D system has a global attractor at $$x=1.1$$, then having this attractor at $$x=1.3$$ would be quantitatively different, but qualitatively the same: you still have a finite global attractor. Now, if suddenly you have two different attractors, or ten, or none, that's a qualitatively change - of the type described by a bifurcation.

The Wikipedia entry the OP already links lists many other examples.