Exact Rayleigh Reduction for Direct Detection of Turning Bifurcations
We introduce an exact Rayleigh reduction for the direct construction and detection of turning bifurcations in nonlinear parameter-dependent equations. A generalized Rayleigh functional ordinarily provides only a scalar necessary constraint on solutions. We show that, along a suitable one-parameter transformation orbit $u=\mathcal G_sϕ$, it can instead become the exact parameter law $λ=\mathcal R(\mathcal G_sϕ)$ of a genuine solution branch. Turning points are then found from nondegenerate critical points of this one-dimensional Rayleigh profile. The branch tangent automatically yields a kernel direction of the state linearization, and under the usual Fredholm, kernel-simplicity, and parameter-transversality assumptions the detected point is a simple fold. For Kirchhoff equations, amplitude scaling on bounded domains and spatial dilation on $\mathbb R^N$ yield explicit exact branches, global parameter thresholds, and countable hierarchies of turning values. For the generalized Kirchhoff law $M_b(A)=a+bA^θ$, the branch geometry is governed by the dimension--homogeneity index $θ(N-2)-2$: an interior turning bifurcation occurs exactly when $θ(N-2)>2$. This gives a universal bifurcation trichotomy that is independent of the particular Berestycki--Lions nonlinearity.