Almost Conjugacy

摘要我们通过考虑这样一个问题来解决科学中与动力系统领域相关的一个广泛的基本建模问题：物理系统的模型什么时候是“好的”表示。虽然共轭提供了一种方法来确定两个系统是否在动态上等价，但没有足够的数学技术来确定“玩具”模型的动力学何时与物理系统的动力学相似(尽管不完全相同)。共轭的概念太过僵化，不能应用于应用动力系统的典型情况，在这种情况下，我可能会说，该模型只是“提醒我们”真实的系统。我们在这里提出了基于不动点迭代格式的方法，以及我们将为更广泛的实际数值应用而开发的一些变体，以产生我们称为“通勤者”的函数。换位器是两个等价动力系统之间的共轭函数，但不同系统之间的坐标变换是非同胚的。这种转换对于动力系统的概念来说是很自然的，因为它在系统的轨道结构语言中匹配系统，而我们的实际计算与符号动力学领域的轨道等价概念有关。我们的关键方法是基于度量这个通勤函数是同胚的失败--我们称之为同胚亏格。中心点是，我们通过量化通勤函数在多大程度上不是同胚来比较不等价的系统，这种方法比基于Banach空间规范的传统比较方法更尊重动力学。在许多给定的科学领域，专家通常不会很难形成对模型质量的看法。例如，心脏病专家可能会同意，某个方程可能会成为一个很好的人体心脏模型，或者气象学家可能会认为，一个特定的低维模拟可能会产生真实的天气。显然，有必要将这种“近似”建模的概念建立在明确的数学基础上，特别是当模型提供对现实世界现象的定性描述时。我们的工作提供了一种计算方法，允许研究人员和科学专家评估哪个模型最适合他们的工作，什么建模参数合适，或者为什么某些模型可能更适合特定的物理情况。这项工作的核心是我们提供了一种在模型中“量化质量”的方法。这项研究直接应用于公共卫生领域的问题，如开发实时心电监测系统以检测异常心率，更好地描述天气模型，提高我们对流体流动系统中湍流的理解，以及更精细地设计结构机械系统，如飞机机翼和桥梁中的那些系统。

AbstractWe address a broadly fundamental modeling issue in science as related to the field of dynamical systems by considering the question: when is a model of a physical system a "good" representation. While conjugacy provides a means to determine whether two systems are dynamically equivalent, there is no sufficient mathematical technology to decide when the dynamics of a "toy" model are like (although not identical to) the dynamics of the physical system. The concept of conjugacy is too rigid and cannot be applied in the typical situation of applied dynamical systems, where me might say that the model only "reminds us" of the true system. We propose here methods based on a fixed point iteration scheme, and some variants which we will develop for broader practical numerical application, to produce a function which we call a "commuter." The commuter is the conjugacy function between two equivalent dynamical systems, but a non-homeomorphic change of coordinates translating between dissimilar systems. This translation is natural to the concepts of dynamical systems in that it matches the systems within the language of their orbit structures, and our practical computation is related to the concept of orbit equivalence from the field of symbolic dynamics. Our key method is based on measuring failure of this commuter function to be a homeomorphism - which we call homeomorphic defect. The central point is that we compare nonequivalent systems by quantifying how much the commuter functions fails to be a homeomorphism, an approach that respects the dynamics better than the traditional methods of comparison based on Banach space norms.Experts in many given fields of science will often have little difficulty in forming opinions of model quality. For example, a cardiac specialist may agree that a certain equation may make a good model of the human heart, or the meteorologist may believe that a particular low-dimensional simulation may produce what "looks like" realistic weather. Clearly there is a need to put this notion of "approximate" modeling on a clear mathematical footing, particularly when the models provide qualitative descriptions of real world phenomena. Our work provides a computational method to allow researchers and scientific experts to assess which model is most suitable for their work, what modeling parameters are appropriate, or why certain models might better fit a certain physical situation. At the heart of the work is that we provide a means to "quantify the quality" in a model. This research has direct application to problems in the public health sphere, such as the development of a real-time EKG monitoring systems to detect abnormal cardiac heart rhythms, to better characterizing models of the weather, to improve our understanding of turbulence in fluid flow systems, and to finer design of structural mechanical systems such as those found in aircraft wings and bridges.