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

无限维动力系统：非线性稳定性、大时间瞬态行为和分岔

• 批准号: 1007450
• 负责人: Margaret Beck
• 金额: $ 14.27万
• 依托单位: Trustees of Boston University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1007450&HistoricalAwards=false
• 关键词: Infinite; dimensional; dynamical; systems; nonlinear

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.

在对物理或生物过程进行建模时，一个主要目标是使用模型来预测在现实世界中将要观察到的系统行为。系统的稳定状态将吸引所有附近的配置，因此它们可以在确定行为中发挥重要作用。另一方面，如果系统向该状态演变的时间尺度很长，那么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，重要的是要了解长期存在的瞬态行为。该建议涉及开发数学技术，以确定给定状态是否稳定，也用于研究这种瞬时行为。此外，分析了一个国家失去稳定性时捕获的某些分叉。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，反应扩散方程中某种类型的时间周期性模式；表征二维Navier-Stokes方程中的瞬态但持久的动力学；通常，分析心脏模型中空间周期性波的稳定性损失及其与不规则心跳的联系。通常，物理和生物过程的数学建模的一个主要目标是预测系统的及时行为。一种方法是寻找所谓的稳定状态。稳定的状态吸引了附近的所有配置，因此可以深入了解将要观察到的行为类型。例如，在研究心脏动力学模型时，定期解决方案可能对应于常规的心跳。如果该解决方案是稳定的，那么即使在其自然节奏中受到干扰，心脏也会放松到常规跳动。但是，如果发生分叉的情况下，周期性解决方案失去稳定性，则可能对应于心律不齐或不规则跳动的情况。由于这种行为是不可取的，因此希望通过数学建模来确定如何调整系统中的某些参数，以使周期解决方案不会失去稳定性。该建议涉及数学技术的发展，这些技术允许人们确定给定解决方案是否稳定，解决方案可能会失去稳定性的方式以及当系统放松到稳定的解决方案时可以观察到的瞬态行为。 该分析是在三个主要的数学环境中进行的，这些设置与生物学的应用有关，例如上述心脏动力学或物理学，例如流体动力学。