I am looking for a reference that state and prove that a bifurcation in a system $F(x, \mu) = 0$ can only appear if $\det(J) = 0$, where $J$ is the Jacobi matrix of $F$ with respect to $x$.

I looked at several books and lecture notes about bifurcation theory, and could find classification of bifurcations assuming it to be true, or mentionning it, but never a direct statement and proof. I assume it is considered textbook material in the field, but I can't find the relevant textbooks.

Also I can prove it myself using the implicit function theorem, but I would really appreciate a nice reference for it instead.

  • $\begingroup$ That statement is not true. For example, the Hopf bifurcation is a counterexample. $\endgroup$ Aug 13, 2022 at 11:58


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