Quantifying complexity and measuring structure within complex systems. Most interesting systems are complex. The complex systems of interest in this project are characterised by a simple dynamical behaviour on each individual part; and a complicated web of interaction between the many distinct parts. The project will focus on the massive system of interacting neurones in the brain, and transmission of influenza via interpersonal contacts. This project will provide a better model of that web of i ....Quantifying complexity and measuring structure within complex systems. Most interesting systems are complex. The complex systems of interest in this project are characterised by a simple dynamical behaviour on each individual part; and a complicated web of interaction between the many distinct parts. The project will focus on the massive system of interacting neurones in the brain, and transmission of influenza via interpersonal contacts. This project will provide a better model of that web of interactions; and new methods for statistically validating this model against data. Existing models of complex networks are statistically biased so, by employing robust statistical methodologies, this problem will be rectified and provide a method for randomly choosing representative complex systems. Read moreRead less
Computer-aided proofs for non-hyperbolic dynamics and blenders . This project aims to develop methods to rigorously detect certain geometric structures in systems that are known to imply chaos and are robust under perturbation. Such structures include blenders and robust heterodimensional cycles and homoclinic tangencies.
This project expects to generate new knowledge in the area of non hyperbolic dynamics utilising a novel combination of recent developments in Dynamical Systems and techniques ....Computer-aided proofs for non-hyperbolic dynamics and blenders . This project aims to develop methods to rigorously detect certain geometric structures in systems that are known to imply chaos and are robust under perturbation. Such structures include blenders and robust heterodimensional cycles and homoclinic tangencies.
This project expects to generate new knowledge in the area of non hyperbolic dynamics utilising a novel combination of recent developments in Dynamical Systems and techniques from rigorous numerics.
Expected outcomes of this project include an efficient computation platform aimed at detecting and verifying chaos-inducing objects in complex dynamical systems.
This should provide significant benefits, such as an increased understanding of non-hyperbolic dynamical systems. Read moreRead less
Discovery and tracking of coherent structures in geophysical flows. Coherent structures in geophysical flows play fundamental roles by organising fluid flow and obstructing transport. For example, ocean eddies strongly influence the transportation of heat, nutrients, phytoplankton, and fish larvae, in both the horizontal and vertical direction. Many coherent structures are very difficult to detect and track by direct measurement (for example satellite observations), and current mathematical tech ....Discovery and tracking of coherent structures in geophysical flows. Coherent structures in geophysical flows play fundamental roles by organising fluid flow and obstructing transport. For example, ocean eddies strongly influence the transportation of heat, nutrients, phytoplankton, and fish larvae, in both the horizontal and vertical direction. Many coherent structures are very difficult to detect and track by direct measurement (for example satellite observations), and current mathematical techniques cannot provide an adequate global description. This project aims to create innovative new mathematical theory and numerical methods to discover and track coherent structures over time frames of physical importance, contributing significantly to our understanding of their role in the oceans' biosphere and climate.Read moreRead less
Innovative mathematics using transfer operators to reveal hidden ordered structures in geophysical flows. Complicated fluid flow is at the heart of physical oceanography and atmospheric science. This research will develop new mathematical technologies to reveal hidden ordered structures around which complicated fluid flow is organised. This new analytical approach will lead to more accurate circulation predictions from ocean and atmosphere models.