Abstract

The local Brouwer degree at a zero  of is given as the sign of the Jacobian,

This assumes nonzero determinant.

Generically, i.e. for in a set of second Baire category, such nondegeneracy holds true automatically.

Indeed, we may simply assume zero to be a regular value of

In our tutorial, we will recall how this simple idea proves homotopy invariance of Brouwer degree for

under a real homotopy (or bifurcation) parameter .

As a corollary, we obtain global bifurcation results for stationary solutions of ODEs

We will also sketch an analogous approach for periodic solutions.

Based on a generic local degree by Alligood, Mallet-Paret and Yorke (1982, 1983), and my 1983 dissertation, we will follow \emph{snakes} of periodic branches to establish global Hopf bifurcation of periodic solutions.

Applied to the motif of feedback cycles, as we have reported earlier, this old method has recently established global Hopf bifurcation in chemical, metabolic, and gene regulatory networks.