Infinite-dimensional dynamical systems: nonlinear stability, large-time transient behaviors, and bifurcation

在对物理或生物过程进行建模时，一个主要目标是使用该模型来预测将在现实世界中观察到的系统行为。系统的稳定状态将吸引所有附近的配置，因此它们可以在决定行为方面发挥重要作用。另一方面，如果系统向该状态演化的时间尺度非常长，人们可能永远无法等待足够长的时间来看到稳定行为的出现。因此，了解持续很长时间的瞬时行为也很重要。这项建议涉及开发数学技术，既可以确定给定状态是否稳定，也可以研究这种暂态行为。此外，还分析了状态失稳时发生的某些分叉现象。这是通过分析三个主要问题来完成的：确定源的非线性稳定性，反应扩散方程中的一种特定的时间周期模式；描述二维Navier-Stokes方程中的瞬时但持续的动力学；分析心脏模型中空间周期波的稳定性损失及其与不规则心跳的联系。物理和生物过程的数学建模的主要目标通常是预测系统在时间上的行为方式。要做到这一点，一种方法是寻找所谓的稳定状态。稳定态吸引了所有附近的构型，因此提供了对将被观察到的行为类型的极大洞察。例如，在研究心脏动力学模型时，周期解可能对应于有规律的心跳。如果这种解决方案是稳定的，那么，即使受到其自然节律的干扰，心脏也会放松到有规律的跳动。然而，如果出现周期解失去稳定性的分叉，这可能对应于将发生心律失常或不规则跳动的情景。由于这种行为是不受欢迎的，人们希望通过数学建模来确定如何调整系统中的某些参数，以使周期解不会失去稳定性。这项建议涉及到数学技术的发展，这些技术允许人们确定给定的解是否稳定，解可能失去稳定的方式，以及当系统松弛到稳定解时可以观察到的暂态行为的类型。分析在三种主要的数学环境中进行，每一种都与生物学中的应用有关，例如上述心脏动力学，或物理学中的应用，例如流体动力学。

When modeling physical or biological processes, one main goal is to use the model to predict the behaviors of the system that will be observed in the real world. Stable states of the system will attract all nearby configurations, and thus they can play an important role in determining the behavior. On the other hand, if the time-scale on which the system evolves toward that state is very long, one may never be able to wait long enough to see the stable behavior emerge. It is therefore important to also understand the transient behaviors that persist for long times. This proposal is concerned with developing mathematical techniques for both determining if a given state is stable and also for investigating such transient behaviors. In addition, certain bifurcations that capture when a state loses stability are analyzed. This is accomplished through the analysis of three main problems: determining the nonlinear stability of sources, a certain type of time-periodic pattern in reaction-diffusion equations; characterizing transient, yet persistent, dynamics in the two-dimensional Navier-Stokes equation; and analyzing the loss of stability of spatially periodic waves in a cardiac model and its connection with irregular heartbeats.One primary goal of the mathematical modeling of physical and biological processes, in general, is to predict the way the system will behave in time. One way to do this is to look for so-called stable states. Stable states attract all nearby configurations, and thus provide great insight into the types of behaviors that will be observed. For example, when studying a model of cardiac dynamics, a periodic solution could correspond to regular heartbeats. If this solution is stable, then, even when subject to disruptions in its natural rhythm, the heart will relax back to regular beating. However, if a bifurcation occurs in which the periodic solution loses stability, this could correspond to a scenario in which arrhythmia, or irregular beating, will occur. Since this behavior is undesirable, one would like to determine, via mathematical modeling, how to adjust certain parameters in the system so that the periodic solution does not lose stability. This proposal concerns the development of mathematical techniques that allow one to determine if a given solution is stable, ways in which the solution could lose stability, and what types of transient behaviors one could observe as the system relaxes to a stable solution. The analysis is carried out in three main mathematical settings, each of which is related to applications in biology, such as the cardiac dynamics mentioned above, or physics, such as fluid dynamics.