Mathematical Sciences: Global Bifurcations of Forced Oscillator Systems

周期性强迫振荡器是非线性数学模型的一个典型例子，用于解释物理、化学和生态系统的过去行为和预测未来行为。研究者继续研究由周期性强迫振荡器系统产生的平面的双参数族地图。广泛的目标是理解这些系统的定性行为，以及当模型中的参数变化时，这些行为是如何变化的，或者“分叉”的。该项目侧重于离散动力系统，而不是微分方程。它旨在将参数空间现象解释为“更自然”的物体在相位和参数空间的笛卡尔积中的投影。该项目包括进一步开发用于“局部”分岔现象的工具，例如周期轨道的诞生，但主要集中在将现有的用于局部现象的工具和结果扩展到用于“全局”现象的类似工具和结果，例如同斜轨道的诞生。研究涉及软件开发，数值模拟，以及数值和全局分岔的数学分析。该项目能够更全面地理解各种分岔问题，包括周期性强迫振荡器。它也为其他离散动力系统族的自动分岔研究提供了工具。研究者继续研究一类用于模拟各种物理、化学和生态系统的系统原型。该项目的目标是研究这些系统在参数改变时的定性行为的变化，或“分支”。例如，颁发的狩猎许可证的数量可以是生态系统数学模型中的一个参数。可能存在一个临界值，低于这个临界值，鹿就能存活，超过这个临界值，鹿就会灭绝。这个临界值称为参数的分岔值。即使是用非常简单的数学公式描述的模型，也可能出现极其复杂的分岔情况。因此，一定程度的数值研究是绝对必要的。该项目包括软件开发、数值模拟以及所涉及的分岔的数值和数学分析。该项目使我们能够更全面地了解各种各样的模型。这可以更好地理解过去，更好地预测像这些模型一样运行的系统的未来，以及许多其他看似不相关的系统。

Peckham A periodically forced oscillator is a prototypical example of a nonlinear mathematical model that is used to explain past behavior and predict future behavior of physical, chemical, and ecological systems. The investigator continues studies of two-parameter families of maps of the plane that are generated by periodically forced oscillator systems. The broad goal is to understand the qualitative behavior of these systems and how that behavior changes, or "bifurcates," as parameters in the models are varied. The project focuses on discrete dynamical systems rather than differential equations. It aims to explain parameter space phenomena as projections of "more natural" objects in the cartesian product of the phase and parameter spaces. The project includes further development of tools for "local" bifurcation phenomena, such as the birth of periodic orbits, but concentrates mostly on extending existing tools and results for local phenomena to analogous tools and results for "global" phenomena, such as the birth of homoclinic orbits. Study involves software development, numerical simulation, and analysis of both the numerics and the mathematics of global bifurcations. The project enables a more complete understanding of a wide variety of bifurcation problems, including periodically forced oscillators. It also provides tools for a more automatic bifurcation study of other families of discrete dynamical systems. The investigator continues studying a prototypical class of systems that is used to model a variety of physical, chemical, and ecological systems. The goal of the project is to study changes, or "bifurcations," in the qualitative behavior of these systems as parameters are changed. For example, the number of hunting licenses issued could be a parameter in a mathematical model of an ecosystem. There might be a critical number of licenses below which deer would survive, and above which the deer would die out. This critical number would be called a bifurcation value of the parameter. An incredibly complex scenario of bifurcations is possible, even for models that are described by very simple mathematical formulas. Hence, some level of numerical investigation is absolutely necessary. The project includes software development, numerical simulation, and analysis of both the numerics and the mathematics of the bifurcations involved. The project enables a more complete understanding of a wide variety of models. This leads to a better understanding the past and a better chance of predicting the future of systems behaving like these models, as well as for many other seemingly unrelated systems.