We introduce the nonlinear generalized Collatz-Wielandt formula λ∗=x∈Qsupi:hi(x)=0minhi(x)gi(x),Q⊂Rn,
Nearby in the stack
and prove that its solution
(x∗,λ∗)
yields the maximal saddle-node bifurcation for systems of equations of the form:
g(x)−λh(x)=0,x∈Q
. Using this we introduce a simply verifiable criterion for the detection of saddle-node bifurcations of a given system of equations. We apply this criterion to prove the existence of the maximal saddle-node bifurcations for finite-difference approximations of nonlinear partial differential equations and for the system of power flow equations.