曲流、零点与Sturm全局吸引子的胞腔结构
Meanders, zero numbers and the cell structure of Sturm global attractors
摘要：
本文研究了半线性偏抛物型微分方程$u_t = u_{xx} + f(x,u,u_x), 0<x<1$的半流的全局吸引子$\mathcal{A}=\mathcal{A}_f$，该方程满足Neumann边界条件.半流的平衡点$v\in\mathcal{E}\subset\mathcal{A}$是偏微分方程的定常解，因此它们是相应的二阶常微分方程边值问题的解。假设所有的平衡点都是双曲的，将$\mathcal{A}$动态分解为不稳定的平衡点流形，给出了Sturm全局吸引子$\mathcal{A}$的几何和拓扑特征，即Sturm复形，其胞元由不稳定的平衡点流形给出.同时，由常微分方程边值问题导出的置换$σ=σ_f$根据平衡点在边界$x=0,1$上的值对平衡点进行排序，完全确定了Sturm全局吸引子$\mathcal{A}$ .等价地，我们使用一条平面曲线，曲流$\mathcal{M}=\mathcal{M}_f$，通过打靶与常微分方程边值问题相联系。本文的主要目的是推导出一个极小极大性质，该性质基于常微分方程边值问题解的Sturm节点性质，直接从置换$σ$，Sturm置换，或等价地从曲流$\mathcal{M}$，Sturm曲流，确定$\mathcal{O}$的单元边界上的平衡点，这些平衡点在边界$x=0,1$处最接近或最远离$\mathcal{O}$。我们强调这一结果的局部方面，通过将其应用到一个例子，其中的平衡在细胞边界的$\mathcal{O}$的识别是从知识的斯特姆曲流$\mathcal{M}$的只有一部分。
英文摘要：
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 $\mathcal{M}$.
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