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

当对物理或生物过程建模时，一个主要目标是使用模型来预测将在现实世界中观察到的系统行为。系统的稳定状态会吸引所有附近的构型，因此它们可以在决定行为方面发挥重要作用。另一方面，如果系统进化到那个状态的时间尺度很长，人们可能永远无法等待足够长的时间来看到稳定的行为出现。因此，理解持续很长时间的瞬态行为也很重要。这一建议是关于发展数学技术，以确定是否一个给定的状态是稳定的，也为研究这种瞬态行为。此外，还分析了在状态失去稳定性时捕获的某些分岔。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，反应扩散方程中的某种类型的时间周期模式；描述二维Navier-Stokes方程中瞬态但持久的动力学；分析了心脏模型中空间周期波稳定性的丧失及其与不规则心跳的关系。一般来说，物理和生物过程的数学建模的一个主要目标是预测系统在时间中的行为方式。一种方法是寻找所谓的稳定状态。稳定的状态吸引所有附近的配置，从而提供了对将被观察到的行为类型的深刻见解。例如，在研究心脏动力学模型时，周期解可以对应于正常的心跳。如果这个溶液是稳定的，那么，即使心脏的自然节律受到干扰，心脏也会放松，恢复正常跳动。然而，如果发生分叉，周期溶液失去稳定性，这可能对应于心律失常或不规则跳动发生的情况。由于这种行为是不可取的，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，使周期解不失去稳定性。这一建议涉及数学技术的发展，使人们能够确定给定的解是否稳定，解可能失去稳定性的方式，以及当系统松弛到稳定解时可以观察到哪些类型的瞬态行为。分析是在三种主要的数学环境中进行的，每一种都与生物学中的应用有关，比如上面提到的心脏动力学，或者与物理学中的应用有关，比如流体动力学。

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.