论文标题
曲折，零数量和Sturm全球吸引子的细胞结构
Meanders, zero numbers and the cell structure of Sturm global attractors
论文作者
论文摘要
我们研究全球吸引力$ \ MATHCAL {a} = \ MATHCAL {a} _f $ semiflows的semiflows的semiflows由半连接的部分抛物线抛物线分化方程式生成的$ u_t = u_t = u_ {xx} + f（x，x，u_x），0 <x <x <x <1 $，满足neumann neumann neumann bource的条件。 Semiflow的平衡$ v \ in \ Mathcal {E} \ subset \ Mathcal {a} $是PDE的固定解决方案，因此它们是相应的二阶Ode边界值问题的解决方案。假设所有平衡的双重性，$ \ Mathcal {a} $的动态分解为不稳定的平衡流形提供了Sturm Global Global吸引子$ \ Mathcal {a a} $作为有限的常规签名的CW-Complexes，与Sturm Complects的细胞相关的几何和拓扑表征，并以不可能的方式散发出了不可或缺的样子。同时，置换$σ=σ_f$分别根据其在边界处订购的值$ x = 0,1 $，从ODE边界值问题衍生而来，完全确定了Sturm Global Global Lafting $ \ Mathcal {a a} $。同等地，我们使用平面曲线，曲折$ \ MATHCAL {M} = \ MATHCAL {M} _F $，通过拍摄与ODE边界值问题关联。 本文的主要目的是在$ \ Mathcal {o} $的单元边界上得出一个最小值的属性，与$ x = 0,1 $在$ x = 0,1 $上最接近或最远离$ \ Mathcal {o} $，直接与Sturm not $ feper $ fecturm nocty $ x $ n permival $ e and cally nath nater cartive n permanty。曲折，基于ODE边界值问题的解决方案的Sturm节点特性。我们通过将其应用于$ \ Mathcal {o} $的单元边界的识别的示例来强调此结果的局部方面，仅根据Sturm曲折$ \ Mathcal {M} $的一部分获得了$ \ Mathcal {O} $的均值。
We study global attractors $\mathcal{A}=\mathcal{A}_f$ of semiflows generated by semilinear partial parabolic differential equations of the form $u_t = u_{xx} + f(x,u,u_x), 0<x<1$, satisfying Neumann boundary conditions. The equilibria $v\in\mathcal{E}\subset\mathcal{A}$ of the semiflow are the stationary solutions of the PDE, hence they are solutions of the corresponding second order ODE boundary value problem. Assuming hyperbolicity of all equilibria, the dynamic decomposition of $\mathcal{A}$ into unstable manifolds of equilibria provides a geometric and topological characterization of Sturm global attractors $\mathcal{A}$ as finite regular signed CW-complexes, the Sturm complexes, with cells given by the unstable manifolds of equilibria. Concurrently, the permutation $σ=σ_f$ derived from the ODE boundary value problem by ordering the equilibria according to their values at the boundaries $x=0,1$, respectively, completely determines the Sturm global attractor $\mathcal{A}$. Equivalently, we use a planar curve, the meander $\mathcal{M}=\mathcal{M}_f$, associated to the the ODE boundary value problem by shooting. The main objective of this paper is to derive a minimax property which identifies the equilibria on the cell boundary of $\mathcal{O}$ which are closest or most distant from $\mathcal{O}$ at the boundaries $x=0,1$, directly from the permutation $σ$, the Sturm permutation, or equivalently from the meander $\mathcal{M}$, the Sturm meander, based on the Sturm nodal properties of the solutions of the ODE boundary value problem. We emphasize the local aspect of this result by applying it to an example for which the identification of the equilibria in the cell boundary of $\mathcal{O}$ is obtained from the knowledge of only a section of the Sturm meander $\mathcal{M}$.
