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Research Theme: Dynamical Systems


Theme Leader: Reinout Quispel (La Trobe)
Deputy Leader: John Roberts (UNSW)
CI: Wolfgang Schief (UNSW), PI Colin Rogers (UNSW).

The complex systems in the flagship applications all involve non-linear dynamical systems.

Goal: To determine the structure and predict the behaviour of non-linear dynamical systems.

Illustrative problem: The transition from micro- to macro-scale models. This will be analysed by two complementary approaches: geometric characterisation of discrete and continuous dynamical systems and stochastic methods for analysing uncertainty.

Major projects confirmed at a recent Theme Workshop held at UNSW on 26 October, 2007 include:
• Data assimilation and forecasting in weather and climate models;
• Mixing phenomena in fluid dynamics;
• Measure preservation and numerical simulation;
• Population dynamics.

Introduction

The theoretical study of dynamical systems started with Hamilton's investigations of systems of many particles, which grew into the area of statistical mechanics. In the modern theoretical approach, the area of dynamical systems has split into two areas, measurable dynamics and continuous dynamics. The former has grown into the complex area of ergodic theory, the latter has connections with partial differential equations via Hamiltonian theory and the theory of evolution equations.

In the former approach, the uncertainty present in a dynamical system is captured by the use of a measure space, and ergodic systems are seen as the basic building blocks. A great deal of the extant theory has concerned the case where time evolution leaves the measure invariant, but recent work by some of the Centre?s Chief Investigators has concerned non-singular systems, where the measure changes with time. New versions of entropy and other invariants have been investigated. While these new perspectives need further theoretical development, it is already clear that they have practical applications. For example, they have been used in coding theory.

Analytical methods such as Bucklund transformations and symmetry analysis have recently been applied by the Centre?s University of NSW researchers to nonlinear physical models. An underlying integrable structure has been revealed in a diversity of application areas of engineering importance such as the theory of fibre-reinforced materials, the deformation of shell membrane structures and toroidal configurations in magnetohydrostatics. The methods adduced, which were geometric in nature, are available through collaborations within the Centre to analyse other nonlinear physical models.

A further example of successful interaction between the disciplines of the Centre has occurred in the use of stochastic differential equations as a tool in partial differential equations and harmonic analysis. The incorporation of knowledge about these stochastic models has led to new insights into various applications, particularly in finance. For example, Brownian motion on the upper half plane has been used to solve the heat equation and hence to price Parisian options and other exotic options. As part of this general theme, the Centre's researchers plan to push this theory further, to promulgate it and to attack new problems from industry. A question which cuts across the other themes of the Centre is that of being able to distinguish properties of stochastic systems from those of deterministic ones, which often appear random. It is possible for the introduction of noise to change system characteristics, sometimes stabilizing systems and sometimes destabilizing them. A classical example of the distinction between these modes of thinking about systems comes from tracking the often dramatic difference in behaviour of a stochastic process and the corresponding mean value process. A cooperative effort from Centre researchers has been used to address this question.

Projects undertaken by the Centre's researchers in this area include:
Dynamical Systems — 2006

TURBULENT UNMIXING
Dynamical Systems Team

A new project on turbulent unmixing has been developed, and industry links are being followed up. The time irreversibility of turbulent motion is usually interpreted to mean that turbulence mixes stuff (fluids, heat, vorticity, grains, reactants, magnetic fields) up, to a degree that is not achievable in realistic times by simple molecular diffusion. That is certainly an important part of the story, but over the past decade there has been increasing recognition that, contrary to expectations, a turbulent flow field can unmix, organise, segregate, cluster, and corral transported quantities.


Dynamical Systems — 2004

MARKOV ODOMETERS
Tony Dooley (Chief Investigator)

The study of non-singular dynamical systems involves a measure space (X,B, _) with either a single transformation T, or a group _ of transformations. Much of the theory that has been developed to date concerns measure-preserving systems, where _ ? _ = _, but many applications demand a weakening of this hypothesis to non-singular systems, where _ ? _ ~ _. A goal of ergodic theory for several decades has been to classify these up to orbit equivalence. A step toward this goal was taken by Dooley and Hamachi (Non-singular dynamical systems, Bratteli diagrams and Markov odometers, Israel J. Math. 38 (2003) 93-123), where it was shown that every such system is orbit equivalent to a Bratteli-Vershik system equipped with a Markov odometer (provided the group _ is amenable). These systems are much easier to calcuate with than the general case: because they are modeled on a particular kind of graph (a Bratteli-Vershik diagram), the transformation can be explicitly described, and the measure is a two-step measure defined by Markov transition matrices. This work has been continued in the thesis of Cruickshank (Vershik systems and non-singular ergodic theory, The University of New South Wales, 2003) who studied the isomorphism classes of non-singular systems using similar methods.The next step in the classification process is to describe which Markov systems are in fact orbit equivalent to product systems. It was an open question for some years as to whether the Morse substitution system is orbit equivalent to a product system. This was shown to be the case by Dooley and Quas, and described in a paper, Approximate transitivity for zero entropy systems?, which has been accepted by the journal Ergodic Theory & Dynamical Systems.

Dynamical Systems — 2004

PAINLEVE STRUCTURE IN ELECTRODIFFUSION
Colin Rogers (Chief Investigator), Wolfgang Schief (Associate Investigator), Kassem Mustaph (Research Fellow)

The theory of electrodiffusion has its origin in work of Nernst and Planck and describes the transport of charged particles through material barriers. In the case of steady state one-dimensional regimes, the Nernst-Planck equation and Gauss law combine to produce on nth order coupled nonlinear system of ODEs where n is the number of distinct ionic charges present. If n = 2 and with ions of equal and opposite changes, the integrable Painleve II equation results. A Backlund transformation for this Painleve II reduction has been previously exploited by the CI and collaborators in an electrolysis model. In recent work, a physically important two-point boundary value problem for the Painleve II equation has been investigated.

Dynamical Systems — 2004

DISCRETE INTEGRABLE SYSTEMS
Reinout Quispel (Chief Investigator)

Most integrable systems have invariants that are ratios of biquadratic polynomials. The systems with an invariant of the latter type belong to the family of QRT mappings (Quispel G. R. W., Roberts J. A. G. and Thompson C J., Physica D 34 (1989) 183). The latter have, in the one-component case, the special quartic polynomial form expressed in terms of 12 parameters of which five correspond to genuine degrees of freedom. The properties of an alternating integrable map whose square is the QRT map has recently been investigated.With H. W. Capel (Netherlands) and J. A. G. Roberts (University of NSW), Quispel has an ongoing project studying duality of discrete integrable systems.

Dynamical Systems — 2005-ongoing

DATA ASSIMILATION
Ian Enting (Professorial Fellow), with Nathan Clisby (RF)


Data assimilation is the process that occurs (at least) daily, as chaotically-evolving models of the atmosphere are re-aligned with observations of the chaotically-evolving real atmosphere, to provide a starting point for a new cycle of weather predictions. Techniques for data assimilation are evolving due to (a) increased computer power allowing more sophisticated approaches; (b) availability of new types of data, especially satellite data; (c) more sophisticated models with more complex processes, especially those involving water and carbon in the land surface.

(2007) Funding from ARCNESS (ARC Network for Earth System Science) supported development of new tools for automatic differentiation.

(2007-ongoing). These techniques are being applied to the development of the CABLE (CSIRO Atmosphere Biosphere Land Exchange) model, the land-surface component (modelling water and carbon) in the new ACCESS (Australian Community Climate and Earth System Simulator) being developed as a collaborative effort between CSIRO, Bureau of Meteorology (and their combined center: CAWCR), along with Australian universities.

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