On a generalized Collatz-Wielandt formula and finding saddle-node bifurcations

03/27/2020

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by   Yavdat Il'yasov, et al.

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We introduce the nonlinear generalized Collatz-Wielandt formula λ^*= sup_x∈ Qmin_i:h_i(x) ≠ 0g_i(x)/ h_i(x),   Q ⊂R^n, and prove that its solution (x^*,λ^*) yields the maximal saddle-node bifurcation for systems of equations of the form: g(x)-λ h(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.