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.