I have a problem that seems similar to the bifurcation problem but that I can't quite seem to mold exactly like that. I have an operator $(I+\lambda A)^{-1}$ acting on a vector $z=x-y$ such that $x$ and $y$ agree for some number of components. For each choice of $\lambda$, there are some number of components of the vector $g=(I+\lambda A)^{-1}z$ such that $g=0$. I want to find the values of $\lambda$ at which the number of $0$ components changes. Since $x$ and $y$ agree in certain places, I believe the number of $0$'s is monotonically decreasing in $\lambda$ with the maximum number of $0$'s in g occuring at $\lambda=0$.
I know very little about bifurcation but this seems sort of like a bifurcation problem (without the derivatives) trying to identify the saddle-node bifurcations. I thought about using power series to write this operator as the derivative of another operator so that I could treat the problem as finding places where the number of equilibria changes.
Does this seems reasonable? If not, can anyone think of a reasonable way to go about this? Thank you so much!