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Given a one-parameter path of equations for which there is a trivial branch of solutions, to determine the points on the branch from which there bifurcate nontrivial solutions, there is the heuristic principle of linearization. That is to say, at each point on the branch, linearize the equation, and justify the inference that points on the branch that are bifurcation points for the path of linearized equations are also bifurcation points for the original path of equations. In quite general circumstances, for the bifurcation of critical points, we show that, at isolated singular points of the path of linearizations, a property of the path that is known to be sufficient to force bifurcation of nontrivial critical points is also necessary. To be more precise, let I be an open interval of real numbers that contains the point λ0 and B an open ball about the origin of a real, separable Hilbert space H. Let ψ : I × B → R be a family of C2 functions. For λ ∈ I, assume ∇xψ(λ, 0) = 0, and set Hessianx ψ(λ, 0) ≡ Lλ. Assume Lλ is invertible if λ = λ0 and Lλ0 is Fredholm. It is known that if the spectral flow of L : I → L(H) across λ0 is nonzero, then in each neighborhood of (λ0, 0) there are pairs (λ, x), x = 0, for which ∇xψ(λ, x) = 0. We prove that if L : I → L(H) is a continuous path of symmetric operators for which Lλ is invertible for λ = λ0, Lλ0 is Fredholm, and the spectral flow of L : I → L(H) across λ0 is zero, then there is an open interval J that contains the point λ0, an open ball B about the origin, and a family ψ : J × B → R of C2 functions such that, for each λ ∈ J, ∇xψ(λ, 0) = 0 and Hessianx ψ(λ, 0) = Lλ, but ∇xψ(λ, x) = 0 if x = 0. Therefore, at an isolated singular point of the path of linearizations of the gradient, under the sole further assumption that the linearization at the singular point is Fredholm, spectral flow is a complete invariant for the detection of bifurcation of nontrivial critical points.

Spectral flow is a complete invariant for detecting bifurcation of critical points