Bifurcation theory and applications

在数学方程组的研究中，分歧理论解释了方程参数的微小变化如何导致方程解的不成比例的变化。 分叉理论告诉我们，这种“因果”关系不是任意的，而最有可能只发生在几个典型的（或一般的）方式，称为分叉，数学家们开始相当好地理解。 在目前的工作中，所研究的方程是非线性微分方程系统，也就是说，方程，确定系统如何随时间演变。 这类方程在科学中有着广泛的应用，有助于理解物理学、工程学和生物学中的许多真实的系统。 这项研究计划有两个目标。 在数学领域，它将有助于分支理论的发展，通过增加新类型的分支知识库。 在数学之外，该研究计划将推进分叉理论思想在现实世界系统中的应用，目的是更好地了解这些系统可能如何变化。 从这个角度来看，目前正在研究的系统的一些例子是：全球气候变化，热交换器阵列（如核反应堆中发现的）的振动模式，人类某些心肺疾病的发病，以及生态学中的可持续生物多样性。

In the study of systems of mathematical equations, Bifurcation Theory explains how small changes in the parameters of the equations can lead to disproportionate changes in the solutions of the equations.  Bifurcation Theory tells us that this "cause and effect" relationship is not arbitrary, but is most likely to occur only in a few typical (or generic) ways, called bifurcations, which mathematicians are beginning to understand fairly well.  In the present work, the equations under study are nonlinear systems of differential equations;  that is, equations that determine how a system evolves with time.   Equations of this kind are widely used in science, and help in the understanding of many kinds of real systems, for example in physics, engineering and biology.  This research program has two objectives.  In the field of mathematics, it will contribute to the growth of Bifurcation Theory, by adding new types of bifurcations to the knowledge base.  Outside of mathematics, this research program will advance the application of ideas of Bifurcation Theory to real-world systems, with the goal of better understanding how these systems are likely to change.  Some examples of the systems currently under investigation from this perspective are:  global climate change, patterns of vibrations in heat exchanger arrays (as found in nuclear reactors), onset of certain cardio-respiratory diseases in humans, and sustainable biodiversity in ecology.