Anticipating Bifurcations for Identifying Dynamic Characteristics of Nonlinear Systems

本研究项目的主要目标是创建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岔以及分岔前和分岔后动力学。复杂系统（从生态系统到工程系统）的动态发生了巨大变化。使用先进的非线性技术预测这类事件是非常重要的。这种复杂系统的行为通常具有非线性特征，可以导致从稳定动力学到不稳定动力学的状态转移或分岔。一种可以定量预测大维非线性系统的分岔以及分岔前和分岔后动力学的方法，将在从纳米系统分析到疾病根除运动的设计等各个领域产生重大影响。三个关键任务是：(1)开发新技术来区分惯性流形动力学和整体动力学，并使用鲁棒信号处理方法处理噪声；(2)开发创新方法来预测分岔图的稳定/不稳定分支；(3)改进应用于复杂非线性系统的一般方法，包括种群动力学和气动弹性系统。这个项目对整个社会有更广泛的影响。这项工作将回答重要的科学问题，并将影响从计算动力学到人口动力学的应用。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，因为这些事件可能导致物种灭绝等不可逆转的后果。如果应用于疾病根除（传染病人群），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，但它可以证明优于大多数其他方法，因为它能够滤除噪声，并且能够在不需要精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.