Noninvertible Dynamical Systems: A Computer-Assisted Study

该研究员研究离散时间系统的动力学和分叉，具有非唯一的逆时间行为。他和他的同事研究二维自同态和三维自同态动力学的各个方面，并开发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：线性化映射变得奇异的点的轨迹。理解临界集的向前和向后迭代是理解不可逆映射完整动力学的关键。 由于时间上向后的轨迹不是唯一的，因此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）自同态的情况扩展到（不可逆）自同态的情况。 这些集和系统牵引器与临界集的相互作用产生了一类重要的动力学现象，不同于那些产生的非线性同态，这种相互作用对参数的依赖性产生了新的分叉。 相应地扩展了当前用于模拟、不变集计算和分叉研究的计算工具，并构造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）的不同原像数量的几何爆炸可能需要新的编程方法，科学可视化在全球动力学及其参数依赖性的数值探索中至关重要。 这个项目涉及软件开发、数值研究和数学分析，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映射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持续的对话，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的定性行为随着参数的变化而发生的变化，或“分叉”。正在研究的这类模型被称为“非可逆离散动力系统”。“离散动力系统是一个提供了一个规则，用于预测未来某个点（比如说，从现在开始的一年）的某个数量或数量，基于该数量现在的值。 通过迭代规则，我们可以预测未来任何一年的数量。当规则可以被“反转”以产生一个规则来追溯时间，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，数值研究和数学分析。 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没有希望理解它们。 另一方面，如果没有相应的数学分析，仅仅依靠计算机研究，通常会导致非常不完整的理解。 因此，这三项活动必须相互配合。 关于这三种活动，以前已经有很多关于一维（thedimension是被监测的数量）可逆和不可逆系统以及更高维可逆系统的工作，但是关于二维和更高维不可逆系统的工作仍然处于相对初期阶段。 这是他们关注的领域。 进步将导致对这类模型的更深入的理解，因此也会导致对它们所模拟的现象的更深入的理解。 在这个过程中，他们开发的软件工具，用于其他人继续研究这些系统，以及科学可视化对象的工具，无论他们来自动力系统应用程序或来自任何科学，工程或数学领域。

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.