Noninvertible Dynamical Systems: A Computer-Assisted Study

研究具有非唯一逆时行为的离散时间系统的动力学和分岔问题。他和他的同事研究了二维自同态和三维自同态动力学的各个方面，并开发了这些研究所需的科学计算、稳定性和可视化工具。研究不可逆映射的基本工具是“临界集”的概念：线性化映射变为奇异的点的焦点。理解criticalset的正向和反向迭代是理解不可逆映射的全部动态的关键。由于时间上向后的轨迹不是唯一的，涉及向后迭代的定义（例如鞍点的不稳定流形）和计算（例如鞍点的稳定流形）必须从（可逆）微分同态情况扩展到（不可逆）自同态情况。这些集和系统吸引子与临界集的相互作用产生了一类重要的动力学现象，这些现象不同于引起非同构的动力学现象，并且这些相互作用对参数的依赖产生了新的分岔。当前用于模拟、不变量集计算和分岔研究的计算工具得到了相应的扩展，并且构建了新的工具来处理多个向后轨迹及其后果。一个点（或一个集合）的不同的预像数量可能的几何爆炸需要新的编程方法，科学可视化在全局动力学及其参数依赖性的数值探索中变得至关重要。他们的项目涉及软件开发、数值调查和数学分析。这三者的相互作用是理解非平凡系统动力学的必要条件。研究者计划对不可逆地图的动态行为进行系统的计算机辅助研究，在计算和理论之间进行连续对话，并考虑到说明性和技术相关的应用。研究者和他的同事继续研究用于模拟各种物理、化学和生态系统的典型方程组的动力学行为。目标是研究这些系统在参数改变时的定性行为的变化，或“分岔”。这类模型被称为“非可逆离散动力系统”。离散动力系统提供了一种规则，可以预测未来某一时刻（例如，从现在起一年后）的某个或多个数量，这种规则仅基于该数量现在的值。通过迭代该规则，我们可以预测未来任何一年的数量。当规则可以被“倒转”以产生一个在时间上向后走的规则时，系统被称为可逆的。否则它是不可逆的。该项目包括软件开发、数值调查和数学分析。这些系统的行为通常是如此复杂，以至于没有计算机的帮助就不可能理解它们。另一方面，单独的计算机调查，没有伴随的数学分析，通常会导致一个非常不完整的理解。因此，这三个活动必须相互协调地进行。关于这三种活动，以前有很多关于一维（维度是被监测的量的数量）可逆和不可逆系统以及高维可逆系统的工作，但是关于二维和高维不可逆系统的工作仍然处于相对初级阶段。这正是他们关注的领域。进步将导致对这类模型的更深入的理解，从而对它们所模拟的现象有更深入的了解。在这个过程中，他们开发了软件工具，这些工具可以用于其他人继续研究这些系统，以及用于科学可视化对象的工具，无论这些工具来自动力系统应用程序还是来自科学、工程或数学的任何领域。

The investigator studies the dynamics and bifurcations ofdiscrete-time systems featuring nonunique reverse-time behavior.He and his colleague investigate two-dimensional endomorphismsand aspects of three-dimensional endomorphism dynamics, anddevelop scientific computation, stability, and visualizationtools necessary for these studies. The basic tool for the studyof noninvertible maps is the notion of a "critical set": thelocus of points where the linearized map becomes singular.Understanding the forward and backward iterates of the criticalset is the key to understanding the full dynamics ofnoninvertible maps. Because trajectories backward in time arenot unique, definitions (e.g. unstable manifolds of saddlepoints) and computations (e.g. stable manifolds of saddle points)involving backward iterates must be extended from the(invertible) diffeomorphism case to the (noninvertible)endomorphism case. The interactions of these sets and the systemattractors with the critical set give rise to a class ofimportant dynamical phenomena distinct from those arising indiffeomorphisms, and the dependence of such interactions onparameters gives rise to new bifurcations. Current computationaltools used for simulation, computation of invariant sets, andstudy of bifurcations are correspondingly extended, and new toolsare constructed to deal with multiple backward trajectories andtheir consequences. The possible geometric explosion of thenumber of distinct preimages of a point (or a set) requires newprogramming approaches, and scientific visualization becomescrucial in numerical exploration of global dynamics and theirparameter dependence. Thr project involves work in softwaredevelopment, numerical investigation, and mathematical analysis.The interplay of all three is necessary to have any hope ofunderstanding the dynamics of nontrivial systems. Theinvestigator plans a systematic computer-assisted study of thedynamic behavior of noninvertible maps, carried out in acontinuous dialogue between computation and theory, and withillustrative and technologically relevant applications in mind. The investigator and his colleague continue studying thedynamical behavior of typical systems of equations that are usedto model a variety of physical, chemical, and ecological systems.The goal is to study changes, or "bifurcations," in thequalitative behavior of these systems as parameters are changed.The class of models being studied is called "noninvertiblediscrete dynamical systems." A discrete dynamical system is onewhich provides a rule for predicting some quantity or quantitiesat some point in the future (say, one year from now) based onlyon the value of that quantity now. By iterating the rule, we canpredict what the quantity will be at any year in the future.When the rule can be "inverted" to produce a rule to go backwardin time, the system is called invertible. Otherwise it isnoninvertible. The project includes software development,numerical investigation, and mathematical analysis. The behaviorof these systems is typically so complicated that there is nohope of understanding them without the aid of computers. On theother hand, computer investigation alone, without the acompanyingmathematical analysis, generally leads to a very incompleteunderstanding. Consequently, all three activities must be donein concert with each other. With respect to all three activites,there has been much previous work on one-dimensional (thedimension is the number of quantities that is being monitored)invertible and noninvertible systems and on higher dimensionalinvertible systems, but work on two-dimensional and highernoninvertible systems is still in relative infancy. It is thisarea on which they focus. Advances will lead to a greaterunderstanding of this class of models, and therefore thephenomena that are modeled by them. In the process, they developsoftware tools that are of use to others for continued study ofthese systems, as well as tools for scientifically visualizingobjects, whether they come from dynamical systems applications orfrom any area of science, engineering or mathematics.