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Holomorphic Solutions for a Class of Functional Differential Equations
by
Bruce van Brunt
Massey University, Palmerston North
In this talk we study special cases of initial-value problems of
the form
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We are concerned with the existence and continuation of nonconstant solutions that are holomorphic at a fixed point. Results from complex dynamics are used to show that, in general, the nonlinear functional term precludes solutions holomorphic at repelling fixed points and produces a natural boundary for holomorphic solutions at the attracting fixed point. For simple cases we investigate the behaviour of solutions near the natural boundary.
Date received: October 29, 2008
Mathematical modelling and parameter identification methods in systems
by
Christopher Eric Hann
Department of Mechanical Engineering, University of Canterbury
Coauthors: J. Geoffrey Chase, Geoffrey M. Shaw, Thomas Desaive, Paul Docherty, Christina Starfinger, Katherine Kok, Richard Brown, Sam Houghton.
The combination of mathematical modelling and parameter identification is a powerful tool for understanding and/or controlling real systems, natural or artificially made. However, modelling and identification are usually viewed as two separate entities and are often in direct conflict with each other. For example as the level of detail in a mathematical model increases, the amount of physical measurements required for validation increases as well as the number of parameters. Therefore, parameter identification can become infeasible with many combinations of parameters providing equally good model fits to the measured data. Parameter identification can also become computationally intractable with excessively large numbers of simulations required to avoid local minima's or false solutions.
This paper takes a minimal modelling approach where only the essential dynamics of a system are captured with the emphasis on the specific application or outcome required. Furthermore, a new concept of parameter identification is presented where for a given differential equation model, the inverse problem and forward problem are treated as one without any requirement of numerical DE solvers. Specifically, the model equations are first reformulated in terms of integrals of measured data. The inverse problem is then transformed into the context of a semi-discrete dynamical system, where the steady state solution corresponds precisely to the "best fit" model solution in the least squares sense. The integral formulation is critical for stability, results in fast convergence, and overall the method is more accurate and 104-106 times faster than current methods, depending on the application.
To demonstrate the concepts, the minimal modelling and parameter identification methodologies are implemented on several case studies in biomedical engineering. However, note, that the methods are general, and could be equally applied in any engineering field, particularly where lumped parameter or compartmental type non-linear differential equation models are used. Examples include the glucose-insulin and cardiovascular systems, with application to diagnosis and therapy in the Christchurch Intensive Care Unit.
A third example concentrates on results of a new technology being developed for non-invasive breast cancer detection. This technology is called Digital Image-based Elasto-tomography or "DIET". Minimal modelling and parameter identification play a major role in both the computer vision side and tissue stiffness reconstruction of the DIET system. Results of the computer vision algorithms applied to silicon breast phantoms are presented. Finally, some initial results of applying the "minimal modelling" approach and integral-based parameter identification to simulated 2D tissue displacement data using the Navier equations are given.
Date received: October 28, 2008
A solvable model for chimera states in heterogeneous networks
by
Carlo Laing
Massey University, Auckland
Chimera states in networks of coupled oscillators occur when some fraction of the oscillators synchronise with one another, while the remaining oscillators are incoherent. Several groups have studied chimerae in networks of identical oscillators, but I will discuss these states in a heterogeneous model for which the natural frequencies of the oscillators are chosen from a distribution. We obtain exact results by reduction to a finite set of differential equations and find that heterogeneity can destroy chimerae, destroy all states except chimerae, or destabilise chimerae in Hopf bifurcations, depending on the form of the heterogeneity.
Date received: October 27, 2008
Through Fourier-tinted spectacles: advection-dispersion equations transformed to a dynamical system
by
Robert McKibbin
Massey University, Albany
A system of partial differential equations that may be used to describe the transport of a pollutant in a groundwater aquifer that is composed of several different parallel sedimentary layers is constructed. The groundwater flow throughout is assumed to be under the influence of a uniform pressure gradient parallel to the layers, and the layer thicknesses are small compared to the lateral extent of the aquifer. Transfer of the pollutant across the layer interfaces may occur when the concentrations within adjoining layers are not equal.
The result is a set of coupled linear time-dependent one-dimensional advection-dispersion equations. A Fourier transform (FT) is applied with respect to the spatial variable; the resulting set of coupled linear first-order ordinary differential equations for the (complex-valued) transformed pollutant concentrations (in time, and carrying the FT parameter) may then be solved using standard linear dynamical system techniques. The predicted pollutant concentrations are retrieved by numerically evaluating the inverse transforms. The complex FT of the concentration in each layer may also be examined using a phase-plane (complex plane). Some illustrative results are presented.
Date received: September 21, 2008
Determining a safe path through a dynamic threat environment in real time
by
G.N. Mercer
UNSW at ADFA, Canberra
Coauthors: H.S. Sidhu and M.P. Rowe
The problem of minimising the risk to a vehicle when travelling through a threat environment is considered. In a military application the threat environment could be a minefield (land or sea) or a radar network. A safest path route is required in real time but often there is limited computational power on the battlefield which restricts the type of methods that can be used. In addition there can be ``pop up'' threats that can dramatically change the safest path with little prior warning. We apply an energy based springs and masses model to determine the safest route. This model reduces down to finding the steady state of a large system of ODEs. The model is very robust and capable of producing good enough real time safe paths even under a rapidly changing threat environment.
Date received: November 6, 2008
A fundamental analysis of a membrane bioreactor containing a sludge disintegration system
by
Mark Nelson
School of Mathematics & Applied Statistics, University of Wollongong
Coauthors: T.C.L. Yue
We analyze the steady-state operation of a membrane bioreactor system (MR) incorporating a sludge disintegration unit (SD). The latter is used to prevent excess sludge production. The relationship between process control parameters and the performance of the MR-SD is determined by finding the steady-states of the model and determining their stability as a function of the residence time. Asymptotic solutions for the steady-state solutions in the limit of high residence times are obtained. These show that at sufficiently high residence times the mixed liquor suspended solids (MLSS) content of the bioreactor is independent of the operation of the sludge disintegration unit. Thus the main role played by the sludge disintegration unit is to improve the performance at `low' residence times. For a specified MLSS concentration the values of the dimensionless residence time and the sludge disintegration factor are determined that ensure zero excess sludge production. If the sludge disintegration factor is sufficiently high then the MLSS content is guaranteed to be below the target value (`negative' excess sludge production) provided that the residence time is higher than the washout value. It is shown that zero excess sludge production can be achieved with a slight decrease in effluent quality.
Date received: October 27, 2008
Multi-scaling analysis of a predator-prey model
by
John J Shepherd
RMIT University
Coauthors: A Stacey, T Grozdanovski
The general Lotka-Volterra equations are the simplest differential equation system for modelling the results of a two-species interaction in which one species is preyed upon by the other. Although, in their simplest form, they are too idealized to accurately model real-world communities, they display features that make them worthy of continued study.
When competition within the species (intraspecies competition) is incorporated, the resulting system moves closer to a realistic representation of real predator-prey populations.
In this talk, we consider such a system, where intraspecies competition occurs in the prey, but not the predator; while the growth rate for the predator is much slower than that for the prey. Such a situation arises in a range of predator-prey communities – for example, foxes and rabbits; lions and antelope.
We exploit this differential in growth rates by applying a multi-scaling technique to obtain approximate expressions for the evolving predator and prey populations, valid over all time. These are then used to describe predator and prey behaviour.
Date received: October 29, 2008
Classical and pulsating combustion waves in a chain-branching reaction model
by
Harvinder Sidhu
Applied and Industrial Research Group, School of Physical, Environmental and Mathematical Sciences, UNSW at ADFA, Canberra
Any scientist who has gazed into a campfire will appreciate the complexity of combustion and the difficulty in constructing a theoretical model of the process. In most investigations of these phenomena only the simplest models have been utilized. These simple models have all featured one-step chemistry, where the reaction is assumed to be well modelled by a single step of fuel and oxidant combining to produce products and heat. In the past these models have been comprehensively analysed using the classical methods of matched asymptotics and linearised stability analysis, yielding now familiar results for planar flame propagation The one-step, large activation energy model has led to many useful qualitatively correct predictions such as: ignition, extinction and stability of diffusion flames, propagation and stability of premixed flames; structure and stability of flame balls. In particular, it has been possible to show many qualitative features of flame stability that can, in a generic sense, be observed in experimental work, but it has not been possible to make general quantitative comparisons to experimental work. This is mainly because real flames do not arise as a result of one-step chemistry. Some researchers also claim that many simple kinetic schemes do not give results which correspond to experimental observations and can produce erroneous conclusions.
At the other extreme, several groups are involved in the study of flame behaviour using full numerical solutions of the equations of energy and mass transfer for all of the species involved with detailed chemistry. Although such investigations are useful in providing quantitative prediction for observed phenomena, there is still a great deal of uncertainty about the reliability of these complex models when applied to the prediction of stable combustion regimes and particularly the onset of exotic combustion phenomena such as pulsating and cellular combustion. It is in these very regimes when the reactions begin to change rapidly in space and/or time, that any numerical method is tested to its extreme limits and the conclusions drawn from these numerical results must be viewed carefully.
The main aim of our current work is to systematically investigate the stability of the flame solutions with complex kinetics. We will progress from the one-step scheme which has been a dominating paradigm in combustion theory, to the two-step scheme and beyond. In today's talk I will present results of our work thus far.
Date received: October 27, 2008
Modelling of coexistence of endophyte-free and endophyte-infected grasses in New Zealand grazing system.
by
Tanya Soboleva
AgResearch Limited, New Zealand
Coauthors: Anthony J. Parsons
While endophyte contributes positively grasslands productivity, it has well-known adverse effects on livestock. New strains of endophytes with desirable properties are cultivated in the laboratory conditions then artificially infected into ryegrasses, which later could be introduced into the pasture.
The presented model is aimed to design best strategies for releasing into environment those ryegrasses artificially infected with new endophytes. The model represents a system of differential equations for two competing types of the ryegrass with different attractiveness to insects and grazers. The system has two non-trivial stable nodes located in distant regions of the phase space. The separatrix is separating the phase trajectories reaching the state with strong prevalence of endophyte-infected grass from the trajectories tending to the state with strong prevalence of endophyte-free grass. Such topology of the phase diagram usually suggests strong sensitivity to initial conditions around the separatrix. However, this is not the case for a feasible (from the point of view of the considered application) time scale. The variables of the system evolve quickly toward some universal relation (a line at the phase diagram) between them and then very slowly approach fixed points. The points along this line, especially those close to the separatrix , can be considered as quasi-steady states for this system. This phenomenon of rapid evolution of dynamical variables toward a universal relationship is the essence of the so-called ``large-river effect'', which was previously studied in the reference to the theory of phase transitions.
Date received: October 20, 2008
An asymptotically stable collision-avoidance system
by
Jito Vanualailai
School of Computing, Information & Mathematical Sciences, University of the South Pacific, Suva, Fiji
Coauthors: Bibhya Sharma and Shin-ichi Nakagiri
Artificial potential fields, which are widely used in robotics for path planning and collision avoidance, are normally beset by difficulties arising from the existence of local minima. This presentation proposes a solution that involves an asymptotically stable point-mass system governed by differential equations. The system represents a planar point robot moving from its initial position to the desired goal whilst avoiding a static obstacle. Because the system is asymptotically stable, its Lyapunov function, which produces artificial potential fields around the goal and the obstacle, has no local minima other than the goal configuration in the pathwise-connected proper subset of free space which contains the goal configuration. As an application, we consider the point stabilization of a planar mobile car-like robot moving in the presence of a static obstacle.
Date received: October 25, 2008
Burning issues: critical storage and assembly.
by
Graeme Wake
Centre for Mathematics in Industry, Massey University, Albany
Coauthors: Weiwei Luo, University of Alabama Huntsville.
The storage problem for solid combustible materials is recast into a dynamical systems framework so as to provide an easily accessible platform for fire investigators to use as a decision-support tool. This is easier than the usual path-following methods and will be the basis for commercially-developed software which is under development. This, and some new work on critical assembly conditions, were developed initially as an aid to a recent marine fire investigation. Some interesting but simple non-local steady-state equations provide useful bounds for the determination of the critical conditions for the storage problem.
Date received: September 15, 2008
A multi-compartment, two population, age-distribution model of cancer cell growth; transfer from in vivo to in vitro
by
David JN Wall
Department of Mathematics & Statistics, University of Canterbury
Coauthors: Liene Daukste and Britta Basse
Human cancers have been shown to contain a population of relatively slower growing cells with a cell cycle time, depending on the tumour, between three days and several weeks. When cells from tumours removed at surgery are grown in culture, they initially grow at this slow rate. However, loss of the slower growing cell population over time results in the emergence of a population of more rapidly growing cells (a cell line) with doubling times of 1-3 days. It has been postulated that the more slowly growing population is maintained by a small population of more rapidly growing population. In this talk we extend a previously developed and analysed model to describe the behaviour of a complex system with two cell populations with different kinetic characteristics. The aim of this model is to provide a framework for understanding the difference in behaviour of cancer cell lines and the human tumours from which they were derived.
We discuss results regarding the stability of age-distributions displaying balanced exponential growth, and then consider the existence of steady age-distributions displaying balanced exponential growth.
Date received: November 3, 2008