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
Discovery Early Career Researcher Award - Grant ID: DE160100147
Funder
Australian Research Council
Funding Amount
$381,294.00
Summary
Coherent structures in chaotic dynamical systems. Using transfer operators and state-of-the-art multiplicative ergodic theory as a springboard, this project aims to develop innovative mathematics for bridging gaps between dynamical systems theory and applications. Coherent structures, such as oceanic eddies and atmospheric vortices, are prevalent in real-world dynamical systems and play a crucial role in both weather and climate systems. These structures arise in externally forced systems, and t ....Coherent structures in chaotic dynamical systems. Using transfer operators and state-of-the-art multiplicative ergodic theory as a springboard, this project aims to develop innovative mathematics for bridging gaps between dynamical systems theory and applications. Coherent structures, such as oceanic eddies and atmospheric vortices, are prevalent in real-world dynamical systems and play a crucial role in both weather and climate systems. These structures arise in externally forced systems, and the existing theory concerning their location, number and stability to model errors is much less understood than in the non-forced counterpart. The intended outcomes include new algorithms for the automatic detection of coherent structures and results about their stability under perturbations which are relevant to roles in both weather and climate systems.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
Spectral Theory of Hamiltonian Dynamical Systems. Stability theory of steady states, travelling waves, periodic waves, and other coherent structures in nonlinear Hamiltonian partial differential equations is a cornerstone of modern dynamical systems. In particular it is of utmost importance to reliably compute eigenvalues, which determine the stability or instability of such structures. This project will develop methods to compute the spectrum of Hamiltonian operators in more than one spatial di ....Spectral Theory of Hamiltonian Dynamical Systems. Stability theory of steady states, travelling waves, periodic waves, and other coherent structures in nonlinear Hamiltonian partial differential equations is a cornerstone of modern dynamical systems. In particular it is of utmost importance to reliably compute eigenvalues, which determine the stability or instability of such structures. This project will develop methods to compute the spectrum of Hamiltonian operators in more than one spatial dimension. It will use the powerful geometric tools of the Maslov index and the Evans function. We will use these to simultaneously advance, and bring together the theories of the two dimensional Euler equations and Jacobi operators.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.
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
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
Geometric methods in mathematical physiology. This project will develop new geometric methods for the analysis of multiple-scales models of physiological rhythms and patterns, and will design diagnostic tools to identify key parameters that cause and control these signals. Thus, this project will deliver powerful mathematics for detecting and understanding fundamental issues of physiological systems.
A coordinate-independent theory for multi-time-scale dynamical systems. Biochemical reaction networks operate inherently on many disparate timescales, and identifying this temporal hierarchy is key to understanding biological behaviour. Currently, the existing dynamical systems theory is not able to rigorously analyse many important biological systems and networks due to this inherent non-standard multi-time-scale splitting. This project aims to remove these stumbling blocks and develop a coordi ....A coordinate-independent theory for multi-time-scale dynamical systems. Biochemical reaction networks operate inherently on many disparate timescales, and identifying this temporal hierarchy is key to understanding biological behaviour. Currently, the existing dynamical systems theory is not able to rigorously analyse many important biological systems and networks due to this inherent non-standard multi-time-scale splitting. This project aims to remove these stumbling blocks and develop a coordinate-independent mathematical theory that weaves together results from geometric singular perturbation theory, differential and algebraic geometry and reaction network theory to decompose and explain the structure in the dynamic hierarchy of events in non-standard multi-time-scale systems and networks.Read moreRead less