论文标题
在广义的Collatz-Wielandt公式上，找到鞍节分叉
On a generalized Collatz-Wielandt formula and finding saddle-node bifurcations
论文作者
论文摘要
我们介绍了非线性通用collatz-wielandt公式$$λ^*= \ sup_ {x \ in q} \ min_ {i：h_i（x）\ neq 0} \ frac {g_i（x）} $（x^*，λ^*）$得出形式方程系统的最大鞍节分叉：$ g（x）-λH（x）= 0，~~ x \ in q $。使用此功能，我们引入了一个简单的可验证标准，用于检测给定方程系统的鞍节分叉。我们应用此标准来证明非线性偏微分方程的有限差差近似值以及功率流程系统的有限差异近似值的存在。
We introduce the nonlinear generalized Collatz-Wielandt formula $$ λ^*= \sup_{x\in Q}\min_{i:h_i(x) \neq 0} \frac{g_i(x)}{ h_i(x)}, ~~Q \subset \mathbb{R}^n,$$ and prove that its solution $(x^*,λ^*)$ yields the maximal saddle-node bifurcation for systems of equations of the form: $g(x)-λh(x)=0, ~~x \in Q$. 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.
