Microcantilevers for multifrequency atomic force microscopy. This project aims to design a microcantilever with high-performing sensors more sensitive and with better noise performance than the typical optical system used in commercial Atomic Force Microscopes (AFMs). The AFM, a nanotechnology instrument, uses a microcantilever (with an extremely shape probe) to interrogate a sample surface. It has made important discoveries in nanotechnology, life sciences, nanomachining, material science and d ....Microcantilevers for multifrequency atomic force microscopy. This project aims to design a microcantilever with high-performing sensors more sensitive and with better noise performance than the typical optical system used in commercial Atomic Force Microscopes (AFMs). The AFM, a nanotechnology instrument, uses a microcantilever (with an extremely shape probe) to interrogate a sample surface. It has made important discoveries in nanotechnology, life sciences, nanomachining, material science and data storage systems. Despite its success, the technique’s spatial resolution and quantitative measurements are limited. This project could lead to breakthrough technologies such as atomic force spectroscopy to study elastic modulus of nanostructures, and establish Australia's prominence in this emerging field.Read moreRead less
What predictions can I trust? Stability of chaotic random dynamical systems. This project aims to make significant progress on the intricate question of global stability of non-autonomous chaotic dynamical systems. Using ergodic theory, this project expects to determine when and how errors in dynamical models that are small and frequent, or large and infrequent, can cause dramatic changes in meaningful mathematical model outputs. Expected outcomes include the discovery of mathematical mechanisms ....What predictions can I trust? Stability of chaotic random dynamical systems. This project aims to make significant progress on the intricate question of global stability of non-autonomous chaotic dynamical systems. Using ergodic theory, this project expects to determine when and how errors in dynamical models that are small and frequent, or large and infrequent, can cause dramatic changes in meaningful mathematical model outputs. Expected outcomes include the discovery of mathematical mechanisms underlying large-scale (in)stability for time-dependent dynamical systems, and reliable numerical methods for detecting instabilities. This research is expected to lead to improved characterisations of shocks or collapse in externally driven dynamical systems and assist scientists to gauge which predictions they can trust.Read moreRead less
Modern mathematics to unravel the birth of coherence in dynamical systems. This project aims to reveal the precise mathematical mechanisms underlying the emergence and disappearance of long-lived coherent features in dynamical systems. This project expects to generate new fundamental mathematics in the area of dynamical systems, using innovative operator-theoretic approaches to carefully tease apart the lifecycles of coherent structures. The expected outcomes of this project include new mathemat ....Modern mathematics to unravel the birth of coherence in dynamical systems. This project aims to reveal the precise mathematical mechanisms underlying the emergence and disappearance of long-lived coherent features in dynamical systems. This project expects to generate new fundamental mathematics in the area of dynamical systems, using innovative operator-theoretic approaches to carefully tease apart the lifecycles of coherent structures. The expected outcomes of this project include new mathematical theory and computational algorithms to anticipate the genesis and destruction of coherent objects, which are key organisers of complex geophysical flows. This breakthrough mathematics should provide significant benefits, such as improved prediction of eddy transport and persistence of weather and climate patterns.Read moreRead less
New mathematics to quantify fluctuations and extremes in dynamical systems. Many problems in the natural world result from the cumulative effect of extreme events in complex dynamical systems. Dynamical models of ecological and physical processes have internal variables that can combine to produce large observable changes. Quantitative estimation of the variability of these chaotic models is difficult because of the time dependence of the dynamics and their “long memory” due to significant deter ....New mathematics to quantify fluctuations and extremes in dynamical systems. Many problems in the natural world result from the cumulative effect of extreme events in complex dynamical systems. Dynamical models of ecological and physical processes have internal variables that can combine to produce large observable changes. Quantitative estimation of the variability of these chaotic models is difficult because of the time dependence of the dynamics and their “long memory” due to significant deterministic components. This project aims to develop mathematics and numerics to accurately quantify and assess these complicated variations. The project expects to provide powerful tools to predict harmful outcomes in biogeophysical systems, and assist with the development of mitigation strategies.Read moreRead less
Extracting macroscopic variables and their dynamics in multiscale systems with metastable states. There are practical barriers to the simulation of complex systems such as molecular systems and the climate system because of the high-dimensionality of the models and the presence of multiscale dynamics. This project will lift these barriers by uncovering the most relevant variables and by creating innovative multiscale simulation algorithms.
Reliable and efficient algorithms for modelling dynamical systems from data. Mathematical and computational models are increasingly important in diverse areas of science and engineering including aircraft and automotive design, robotics, medical sensing, and biology. However, finding an accurate model remains a difficult task. This project will develop new methods to reliably find highly accurate models from recorded data.
A New Approach to High-Performance Control of Nonlinear Systems. The coming generation of robots are highly mobile and will interact significantly with their environment, each other, and human collaborators. However, this leads to highly coupled nonlinear dynamical behaviour, and achieving accurate and reliable control of these systems is pushing current control theory to breaking point. This project aims to develop a new approach to control of nonlinear systems based on contraction theory and c ....A New Approach to High-Performance Control of Nonlinear Systems. The coming generation of robots are highly mobile and will interact significantly with their environment, each other, and human collaborators. However, this leads to highly coupled nonlinear dynamical behaviour, and achieving accurate and reliable control of these systems is pushing current control theory to breaking point. This project aims to develop a new approach to control of nonlinear systems based on contraction theory and convex optimisation, extending the power of optimisation-based control from linear to non-linear systems. The project is expected to lead to new theoretical developments, constructive algorithms and software, and experimental demonstrations on a range of platforms including bipedal walking robots and underwater robots.Read moreRead less
Dynamics of atherosclerotic plaque formation, growth and regression. This project aims to provide a mathematical framework to interpret plaque growth. Many biological processes contribute to the growth of atherosclerotic plaques inside arteries. Lipoproteins enter the artery walls and stimulate tissues to signal to cells which duly respond so that fatty streaks form and grow into dangerous plaques that cause heart attacks or stroke. These processes are often nonlinear and operate on widely varyi ....Dynamics of atherosclerotic plaque formation, growth and regression. This project aims to provide a mathematical framework to interpret plaque growth. Many biological processes contribute to the growth of atherosclerotic plaques inside arteries. Lipoproteins enter the artery walls and stimulate tissues to signal to cells which duly respond so that fatty streaks form and grow into dangerous plaques that cause heart attacks or stroke. These processes are often nonlinear and operate on widely varying time scales. The project plans to use systems of ordinary differential equations, partial differential equations with non-standard boundary conditions, and bifurcation theory to find how nonlinear processes shape plaque growth. The expected results may demonstrate the importance of bifurcations, dynamics and nonlinear systems in plaque growth and provide new models to interpret biological data.Read moreRead less
A geometric theory for non-standard relaxation oscillators. This project aims to develop new geometric methods for the analysis of multi-scale models of biological rhythms, and design diagnostic tools to identify key parameters that cause and control these signals. Rhythms, such as breathing, neural and cardiac rhythms and pulsatile hormone secretion, are central for life. Many important biochemical cell signals exhibiting relaxation-type behaviour cannot be rigorously analysed with standard dy ....A geometric theory for non-standard relaxation oscillators. This project aims to develop new geometric methods for the analysis of multi-scale models of biological rhythms, and design diagnostic tools to identify key parameters that cause and control these signals. Rhythms, such as breathing, neural and cardiac rhythms and pulsatile hormone secretion, are central for life. Many important biochemical cell signals exhibiting relaxation-type behaviour cannot be rigorously analysed with standard dynamical systems tools due to an inherent non-uniform time-scale splitting in these models. This project aims to develop a unified mathematical theory that weaves together results from geometric singular perturbation theory and algebraic geometry to explain the genesis of complex rhythms and patterns in biological, non-standard, multi-scale systems, both at individual and network level.Read moreRead less
Mathematical model reduction for complex networks. This project aims to develop new mathematical methodology to describe the collective behaviour of large networks of oscillators with parameters called collective coordinates. This will allow for the quantitative description of finite-size networks as well as chaotic dynamics, which are both out of reach for current model reduction methods. The project will apply methodology to understand the causes of, and ways to prevent, glitches and failure i ....Mathematical model reduction for complex networks. This project aims to develop new mathematical methodology to describe the collective behaviour of large networks of oscillators with parameters called collective coordinates. This will allow for the quantitative description of finite-size networks as well as chaotic dynamics, which are both out of reach for current model reduction methods. The project will apply methodology to understand the causes of, and ways to prevent, glitches and failure in the emerging modern decentralised power grids. This will develop a framework to address this question, tailored to deal with the hitherto uncharted case of finite-size networks.Read moreRead less