國立清華大學 數學系

Abstract：
This presentation introduces a unified framework for forecasting, detecting, and explaining critical transitions in high-dimensional real-world systems. We begin with a theoretically grounded early-warning signal (EWS), the Dynamical Eigenvalue (DEV), which anticipates both the timing and the type of impending bifurcations. To address high dimensionality, we extend DEV to the Manifold-projected Multivariate DEV (MM-DEV), which integrates nonlinear manifold learning to deliver robust pre-transition warnings, as demonstrated in controlled microbial ecosystems and long-term lake datasets. For retrospective detection of real-world bifurcation, we present Nested Library Analysis (NLA), an equation-free method that accurately identifies the timing of past regime shifts even if the transition signals were hidden within high-dimensional complex dynamics. Finally, we examine the mechanistic drivers of collapse through a high-resolution study of a methanogenic ecosystem, showing that system-wide failure arises from a cascade of localized transitions that dismantle critical network hubs and metabolic pathways. In brief, these algorithms form a comprehensive toolkit, spanning theoretical EWS development, scalable multivariate forecasting, retrospective detection, and mechanistic dissection, offering new insight into how bifurcations unfold in complex systems.