We 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.
