Bifurcation theory and applications

在数学方程系统的研究中，分支理论解释了方程参数的微小变化如何导致方程的解的不成比例的变化。分支理论告诉我们，这种因果关系不是任意的，而是最有可能只以几种典型的(或一般的)方式发生，称为分支，数学家们开始很好地理解这一点。在目前的工作中，所研究的方程是非线性微分方程组；也就是说，决定一个系统如何随时间演化的方程。这类方程在科学中得到了广泛的应用，并有助于理解许多类型的真实系统，例如在物理、工程和生物中。这个研究计划有两个目标。在数学领域，它将通过在知识库中添加新类型的分叉来促进分叉理论的发展。除了数学之外，这个研究计划将推进分叉理论的思想在现实世界系统中的应用，为了更好地了解这些系统可能发生的变化，目前正在从这个角度研究的系统的一些例子包括：全球气候变化、热交换器阵列中的振动模式(如在核反应堆中发现的那样)、人类某些心肺疾病的发病以及生态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.