Statistical Properties of Physical Chaotic Flows

This project concerns the study of statistical aspects of flows, in particular those flows arising from the Lorenz system and various mechanical models. The mathematical concept of a flow is a system which evolves according to some predetermined rule and that this evolution happens in continuous time. This project involves the study of flows which are chaotic of nature in the sense that they have sensitive dependence on initial conditions, sometimes known as the butterfly effect. In other words, knowledge about the system is quickly lost. We focus on the rigorous mathematical understanding of such systems. It is an important topic in the mathematical field of research known as dynamical systems. The sensitivity to initial conditions means that it is neither possible nor useful to study the evolution of individual trajectories. Instead one must make the connection with probability theory and study the statistical properties of the system. For example it is important to understand the rate at which initial information about the system is lost. From another point of view, this is an understanding of the extent to which subsequent observations behave like independent random events. The field of dynamical systems has seen massive advances in recent decades, particularly in the improvement of the mathematical machinery for studying the systems. Initially the systems studied were idealised and far from being real physical examples. The technological progress means that now it is feasible to study these systems of realistic physical character. Finally we can obtain substantial results concerning the systems which were a major motivation behind the development of the field of dynamical systems.

The project focussed on the study of statistical aspects of flows, in particular those flows arising from the Lorenz system and various mechanical models. The mathematical concept of a flow is a system which evolves according to some predetermined rule and that this evolution happens in continuous time. This project involves the study of flows which are chaotic of nature in the sense that they have sensitive dependence on initial conditions, sometimes known as the butterfly effect. In other words, knowledge about the system is quickly lost. The aim was a rigorous mathematical understanding of such systems. It is an important topic in the mathematical field of research known as dynamical systems.The sensitivity to initial conditions means that it is neither possible nor useful to study the evolution of individual trajectories. Instead one must make the connection with probability theory and study the statistical properties of the system. For example it is important to understand the rate at which initial information about the system is lost. From another point of view, this is an understanding of the extent to which subsequent observations behave like independent random events.The field of dynamical systems has seen massive advances in recent decades, particularly in the improvement of the mathematical machinery for studying the systems. Initially the systems studied were idealised and far from being real physical examples. The technological progress means that now it is feasible to study these systems of realistic physical character. Finally we can obtain substantial results concerning the systems which were a major motivation behind the development of the field of dynamical systems.The project was successful in improving our knowledge of these matters in some particular examples of the type described above. Also progress was made in improving the mathematical machinery available to us for investigating the phenomena involved with these systems.