Anticipating Bifurcations for Identifying Dynamic Characteristics of Nonlinear Systems

该研究项目的主要目的是建立一种新的方法来定量预测具有低维惯性流形的大维非线性系统的分叉以及分叉前后的动力学。从生态系统到工程系统，复杂系统的动力学发生了戏剧性的变化。利用先进的非线性技术对这类事件进行预报具有重要意义。这类复杂系统的行为通常以非线性为特征，这可能导致区域从稳定动力学到不稳定动力学的转变或分叉。一种能够定量预测大维非线性系统的分叉以及分叉前后动力学的方法将在从纳米系统的分析到疾病根除运动的设计等多个领域产生重大影响。这三个关键任务是：(1)开发新的技术来区分沿惯性流形的动力学和整体动力学，并使用稳健的信号处理方法来处理噪声；(2)开发新的方法来预测分岔图的稳定/不稳定分支；以及(3)改进应用于包括种群动力学和气动弹性系统在内的复杂非线性系统的一般方法。这个项目对整个社会有更广泛的影响。这一努力将回答重要的科学问题，并将影响从计算动力学到种群动力学的各种应用。例如，迫切需要可靠的方法来预测植物和/或动物种群中的灾难性事件，因为这种事件可能导致物种灭绝等不可逆转的后果。当应用于根除疾病(传染病人群)时，这种方法的潜在影响甚至更大。虽然疾病动力学是一个非常复杂的系统，该方法可能并不完美，但事实证明，它的表现优于大多数其他方法，因为它能够过滤噪音，并能够在不需要准确模型的情况下提供预测。

The main goal of this research project is to create a novel method to quantitatively forecast bifurcations as well as the pre- and post-bifurcation dynamics of large dimensional nonlinear systems with a low dimensional inertial manifold. Dramatic changes in the dynamics of complex systems, from ecosystems to engineered systems, occur. Forecasting such events using advanced nonlinear techniques is of major importance. The behavior of such complex systems is commonly characterized by nonlinearities that can lead to regime shifts or bifurcations from a stable to an unstable dynamics. A method that can quantitatively predict bifurcations as well as the pre- and post-bifurcation dynamics for large dimensional nonlinear systems would have a significant impact in a variety of fields, from the analysis of nano-systems to the design of disease eradication campaigns. The three key tasks are to: (1) develop novel techniques to differentiate the dynamics along the inertial manifold from the overall dynamics and to handle noise using a robust signal processing methodology, (2) develop innovative methods to forecast stable/unstable branches of bifurcation diagrams, and (3) refine the general methods for application to complex nonlinear systems including population dynamics and aeroelastic systems. This project has broader impacts on the society at large. This effort will answer important scientific questions, and will impact applications spanning from computational dynamics to population dynamics. For example, there is an acute need for reliable methods to predict catastrophic events in populations of plants and/or animals because such events can lead to irreversible consequences such as extinction of species. The potential impact of this method is even higher when applied to disease eradication (populations of infectious diseases). While the dynamics of diseases is a very complex system and the method may not be perfect, it can prove to outperform most other methods because of its ability to filter out noise and the ability to provide forecasts without the need for an accurate model.