Exact Rayleigh Reduction for Direct Detection of Turning Bifurcations
Abstract
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=𝒢sϕ, it can instead become the exact parameter law λ=ℛ(𝒢sϕ) 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 ℝN yield explicit exact branches, global parameter thresholds, and countable hierarchies of turning values. For the generalized Kirchhoff law Mb(A)=a+bAθ, 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.
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BibTeX
@misc{ilyasov2026exact,
title = {{Exact Rayleigh Reduction for Direct Detection of Turning Bifurcations}},
author = {Yavdat Il'yasov},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29150},
eprint = {2608.29150},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29150},
}
RIS
TY - UNPB
TI - Exact Rayleigh Reduction for Direct Detection of Turning Bifurcations
AU - Yavdat Il'yasov
PY - 2026
DA - 2026/08/29
PB - arXiv
DO - 10.48550/arxiv.2608.29150
UR - https://arxiv.org/abs/2608.29150
AB - 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=𝒢_sϕ, it can instead become the exact parameter law λ=ℛ(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 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.
ER -