Mathematical Sciences: Global Bifurcations of Forced Oscillator Systems

佩卡姆 周期性强迫振荡器是一个典型的非线性数学模型，用于解释过去的行为和预测未来的行为的物理，化学和生态系统。 调查员继续研究的两个参数家庭的地图所产生的定期被迫振荡器系统的平面。 广义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，这种行为如何变化或“分叉”。 该项目侧重于离散动力系统，而不是微分方程。 它的目的是将参数空间现象解释为“更自然”物体在相空间和参数空间的笛卡儿积中的投影。 该项目包括进一步开发“局部”分叉现象的工具，例如周期轨道的诞生，但主要集中于将局部现象的现有工具和结果扩展到“全局”现象的类似工具和结果，例如同宿轨道的诞生。 研究涉及软件开发，数值模拟，和分析的数值和数学的全球分叉。 该项目能够更全面地了解各种各样的分叉问题，包括周期性强迫振荡器。 它也为其他离散动力系统族的更自动的分岔研究提供了工具。 研究人员继续研究一类原型系统，用于模拟各种物理，化学和生态系统。 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“分叉”。 例如，发放的狩猎许可证数量可能是生态系统数学模型中的一个参数。 可能有一个临界数量的许可证，低于这个数量的鹿可以生存，超过这个数量的鹿就会灭绝。 这个临界值被称为参数的分叉值。 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叉情况。 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，以及对所涉及的分叉的数值和数学分析。 该项目可以更全面地了解各种模型。 这导致更好地理解过去，并更好地预测行为类似这些模型的系统的未来，以及许多其他看似无关的系统。

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.