A simple example I considered is three identical un-coupled ODEs
$\dot{x}_j = r x_j - x^3_j, \quad r,x_j \in \mathbb{R}, \quad j = 1,2,3.$
When $r>0$, there are multiple fixed points with different indices in the state space $\mathbb{R}^3$. If I define an index of a fixed point as
$I = (-1)^{N_r}$
where $N_r$ is the number of repelling eigen-directions of that fixed point, then in this case they are: One unstable node (+1), eight stable nodes (+8), and eighteen saddle points (6-12=-6). So the total index is +3.
However, when $r\le 0$, through pitch-fork bifurcations, there is only one stable node at the origin $(x_1, x_2, x_3) = (0,0,0)$, which has index +1.
Should the total index be conserved through a bifurcation? Thanks!