发表机构
Institute of Mathematics with Computing Centre, Ufa Federal Research Centre of the Russian Academy of Sciences（俄罗斯科学院乌法联邦研究中心数学与计算中心研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出精确瑞利约化方法，用于检测依赖参数的非线性方程的转向分岔，针对基尔霍夫方程得到显式分支等结果，给出与非线性项无关的普适分岔三分法。
AI 中文摘要
我们提出一种精确瑞利约化方法，用于直接构造和检测依赖参数的非线性方程中的转向分岔。广义瑞利泛函通常仅对解提供标量必要约束，我们证明，沿合适的单参数变换轨道$u=\boldsymbol{\text{G}}_s\boldsymbol{\text{\text{φ}}}$，它可转化为真实解分支的精确参数律$\boldsymbol{\text{λ}}=\boldsymbol{\text{R}}(\boldsymbol{\text{G}}_s\boldsymbol{\text{\text{φ}}})$。转向点可从该一维瑞利剖面的非退化临界点中找到，分支切线自动给出状态线性化的核方向，在通常的弗雷德霍姆、核简单性及参数横截性假设下，检测到的点为简单折叠。对于基尔霍夫方程，有界域上的振幅标度和$\boldsymbol{\text{R}}^N$上的空间膨胀可得到显式精确分支、全局参数阈值及可数的转向值层级。对于广义基尔霍夫定律$\boldsymbol{\text{M}}_b(\boldsymbol{\text{A}})=\boldsymbol{\text{a}}+\boldsymbol{\text{b}}\boldsymbol{\text{A}}^\boldsymbol{\text{θ}}$，分支几何由维度齐次指数$\boldsymbol{\text{θ}}(\boldsymbol{\text{N}}-2)-2$控制：当$\boldsymbol{\text{θ}}(\boldsymbol{\text{N}}-2)>2$时，恰好发生内部转向分岔，这给出了与特定Berestycki-Lions非线性项无关的普适分岔三分法。
英文摘要
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.
Comments32 pages, 2 figures
