Anomalous Phase Transitions in Complex Systems

Sudden qualitative changes in the behavior of large complex systems can both be essential for their functioning and also lead to dramatic breakdown. Examples range from ecosystems with possible tipping points and extinction, to sudden changes in heart rhythm, and to self-organization in early development. Mathematical theory aims at predicting such changes, finding mechanisms that can prevent dramatic outcomes, or identifying causes of malfunction. Much of the theory developed in the mathematical sciences addresses transitions in simple dynamical systems, leading to universal qualitative predictions for qualitative changes in behavior. This project is concerned with case studies of transitions in large and complex systems, in particular with situations where such systems behave in apparently anomalous ways, challenging our understanding inferred from existing theory. Examples show how the complexity of large systems can eliminate system memory in transitions, thus facilitating easy switching between qualitatively different states with implications in systems biology. Transitions also involve intricate interplay between spatial organization and temporal evolution, with implications for the formation of ecological clusters and niches, as well as for the emergence of spiral waves in cardiac arrhythmia. The project integrates work of several graduate students on the projects and offers opportunities for mentoring by graduate students and research by undergraduate students during a summer project. This project focuses on the analysis of bifurcations in complex systems that mediate phase transitions. The project aims to identify coherent structures that organize the dynamics, as pacemakers, or as crystalline phases, and study their instabilities as well as the impact of spatial heterogeneity. Specifically, the project investigates interacting particle systems with competing attractive and repulsive forces, which exhibit a striking reversible phase transition from a perfectly mixed state to sorted states and states with clusters and vacuum regions. The anchoring of spiral waves at impurities and a geometric instability leading to bending of spiral arms will also be investigated. Lastly, the project studies striped patterns and the impact of spatial heterogeneity, with a focus on selection mechanisms, for position, wavenumbers, and orientation. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.