Purdue University, USA.
IEEE Trans Vis Comput Graph. 2011 Dec;17(12):1765-74. doi: 10.1109/TVCG.2011.254.
Area-preserving maps are found across a wide range of scientific and engineering problems. Their study is made challenging by the significant computational effort typically required for their inspection but more fundamentally by the fractal complexity of salient structures. The visual inspection of these maps reveals a remarkable topological picture consisting of fixed (or periodic) points embedded in so-called island chains, invariant manifolds, and regions of ergodic behavior. This paper is concerned with the effective visualization and precise topological analysis of area-preserving maps with two degrees of freedom from numerical or analytical data. Specifically, a method is presented for the automatic extraction and characterization of fixed points and the computation of their invariant manifolds, also known as separatrices, to yield a complete picture of the structures present within the scale and complexity bounds selected by the user. This general approach offers a significant improvement over the visual representations that are so far available for area-preserving maps. The technique is demonstrated on a numerical simulation of magnetic confinement in a fusion reactor.
保面积映射存在于广泛的科学和工程问题中。它们的研究具有挑战性，因为通常需要大量的计算资源来检查它们，更根本的是因为突出结构的分形复杂性。对这些地图的视觉检查揭示了一个引人注目的拓扑图像，其中包含嵌入在所谓的岛链、不变流形和遍历行为区域中的固定（或周期）点。本文关注的是从数值或分析数据中具有两个自由度的保面积映射的有效可视化和精确拓扑分析。具体来说，提出了一种自动提取和特征化固定点并计算它们的不变流形（也称为分隔线）的方法，以在用户选择的尺度和复杂度范围内生成存在结构的完整图像。与迄今为止可用的保面积映射的视觉表示相比，这种通用方法有了显著的改进。该技术在聚变反应堆中磁约束的数值模拟上得到了验证。
