Seminars

11 October 2012
Stochastic Regular Grazing Bifurcations
Dr David Simpson (Massey University, Palmerston North)
1 pm
303S-561

Nonsmoothness and noise are two features of a dynamical system that may be the cause of important qualitative behaviour. Hybrid and piecewise-smooth systems are utilized in a wide variety of fields to model phenomena that involve switching, impacts or other nonsmooth elements. In addition, noise may have a variety of interesting effects such as to suppress chaos or induce regular oscillations in a quiescent system. In this talk I will describe the effects of incorporating noise into a canonical vibro-impacting model which has been applied to atomic force microscopy, gear assemblies, metal cutters, vibrating heat-exchanger tubes and other mechanical systems. An attractor of the deterministic model becomes an invariant density when noise is added. Under parameter change, the density transitions between approximately Gaussian and highly non-Gaussian forms. Impacts correspond to an extreme nonlinearity and as a consequence the size of the invariant density may be proportional to the square-root of the noise amplitude.

4 October 2012
NON-DECAYING INTERACTIONS BETWEEN SQUIRMING MICRO-ORGANISMS
Dr Richard Clarke (Auckland University)
1 pm
303S-561

There has been much recent interest in the use of certain prototype microscopic swimmers, called Squirmers, to theoretically explain the involved behaviour observed when micro-organisms such as Opalina or Volvox swim in semi-dilute suspensions and\or near surfaces. Squirmer models aim to emulate, in the most mathematically-tractable way, the means by which such micro-organisms propel themselves through fluid via a syncronised beating of their surface coating of cilia. Indeed it has been shown that the simple dynamical systems derived using two-dimensional Squirmers can successfully reproduce the complex near-wall behaviour seen in more advanced simulations and experiments (Crowdy, 2010). We use this framework to consider the pairwise interaction of such micro-organisms, and show that in two-dimensions the hydrodynamic interaction between squirmers is non-negligible even at asymptotically-large distances between the swimmers, and we discuss the subsequent ramifications for the collective dynamics of squirmer colonies.

27 September 2012
Canard theory and neuronal dynamics
Professor Martin Wechselberger (University of Sydney)
1 pm
303S-561

An important feature of most physiological systems is that they evolve on multiple time-scales. It is the interplay of the dynamics on these different scales that creates complicated rhythms and patterns. In conjunction with the innovative blow-up technique, geometric singular perturbation theory delivers rigorous results on pattern generation in such multiple time-scale problems. As a case study of geometric singular perturbation theory, I will focus on a single neuron model by McCarthy et al. (2008) that looks at the effect of the anesthetic propofol on such a neuron. It is well known that propofol causes paradoxical excitation in low doses. I will show that "canards", exceptional solutions in singular perturbation problems which occur on boundaries of regions corresponding to different dynamic behaviors, provide a possible explanation of the observed paradox. This is joint work with M. McCarthy, N. Kopell (Boston University) and J. Mitry (University of Sydney).

20 September 2012
Electrical tomography imaging in pharmaceutical processes
Dr. Ville Rimpilainen (Auckland University)
1 pm
303S-561

Electrical capacitance tomography (ECT) is an electrical imaging modality that can be used to estimate electrical properties inside different processing vessels. The presentation will describe how ECT has been applied in monitoring of two common pharmaceutical unit processes: high-shear granulation and fluidized-bed drying. In high-shear granulation pharmaceutical powders are built-up into granules with the help of high-shear mixer and binder liquid. The wet granules are usually dried in a fluidized-bed dryer with the help of a heated stream of air. In the presentation it is shown how ECT is used to generate monitoring signals that describe the progress and the characteristics of the processes.

13 September 2012
STABLE LEAVES, PROBABILITY AND FALLING INTO HOLES
Dr. Rua Murray (University of Canterbury)
1 pm
303S-561

Nonlinear dynamical systems, such as those given by the famous Lorenz equations, can exhibit a startling array of interesting behaviour. It is now well-known that the Lorenz equations are chaotic (from both topological, and probabilistic viewpoints) at large ranges of parameter values. Indeed, for significant chunks of parameter space, the chaos is ``invisible'' to all but the most sophisticated mathematical techniques. This talk will survey some of the ways in which an ``ergodic'' viewpoint can describe what is going on. The tour will include an explanation of what the words in the title have to do with the Lorenz equations!

23 August 2012
A Primer to Periodic Problems in Plates: Crystals and Clusters
Mike Smith (The University of Auckland)
1 pm
303S-561

In this talk I will discuss an emerging sub-field in the study of photonic crystals known as platonics. In particular, I will give an introduction to platonic crystals, which are novel structures designed for steering and dispersing flexural wave energy through elastic plates.

16 August 2012
Why PDEs is so different from ODEs: from analytic and numerical viewpoints
Dr Shixiao Wang (Auckland University)
1 pm
303S-561

I will show some existence results of ODEs and PDEs. The aim is to reexamine the fundamental difference between ODEs and PDEs from both analytic and numerical viewpoints. I will also present a novel, unified approach to the numerical methods for solving PDEs, which has recently been developed by Angela Tsai, Robert Chan and myself.

10 August 2012
The numerical computation of violent liquid motion
Professor Frederic Dias (University College Dublin)
3 pm
303S-561

Liquid impact is a key issue in various industrial applications (seawalls, offshore structures, breakwaters, sloshing in tanks of liquefied natural gas vessels, wave energy converters, offshore wind turbines, etc). Numerical simulations dealing with these applications have been performed by many groups, using various types of numerical methods. In terms of the numerical results, the outcome is often impressive, but the question remains of how relevant these results are when it comes to determining impact pressures. The numerical models are too simplified to reproduce the high variability of the measured pressures. In fact, for the time being, it is not possible to simulate accurately both global and local effects. Unfortunately it appears that local effects predominate over global effects when the behaviour of pressures is considered. Having said this, it is important to point out that numerical studies can be quite useful to perform sensitivity analyses in idealized conditions such as a liquid mass falling under gravity on top of a horizontal wall and then spreading along the lateral sides. Simple analytical models inspired by numerical results on idealized problems can also be useful to predict trends. The talk is organized as follows: After an introduction on some of the industrial applications, it will be explained to what extent numerical studies can be used to improve our understanding of impact pressures. Results on a liquid mass hitting a wall obtained by various numerical codes will be shown.

2 August 2012
Gravitational waves: General properties and numerical simulation
Professor Joerg Frauendiener (University of Otago)
1 pm
303S-561

Gravitational waves have been predicted by Einstein in 1917 based on his theory of general relativity. In this talk we will present some of their basic properties such as the sources for gravitational waves and a brief discussion of detectors. The main focus of the presentation will lie on the mathematical theory of the waves and our attempts to model them numerically.

2 August 2012
Gravitational waves: General properties and numerical simulation
Professor Joerg Frauendiener (University of Otago)
1 pm
303S-561

Gravitational waves have been predicted by Einstein in 1917 based on his theory of general relativity. In this talk we will present some of their basic properties such as the sources for gravitational waves and a brief discussion of detectors. The main focus of the presentation will lie on the mathematical theory of the waves and our attempts to model them numerically

26 July 2012
A MODEL VALIDATION STRATEGY APPLIED ON A CONSTITUTIVE VISCOELASTIC MODEL
Professor Castello (Federal University of Rio de Janeiro)
1 pm
303S-561

In this talk it will presented a strategy to determine the level of confidence of a model used to describe viscoelastic behavior. This strategy encompasses model calibration and the assessment of its predictive capabilities in different experimental scenarios.

23 July 2012
A MODEL VALIDATION STRATEGY APPLIED ON A CONSTITUTIVE VISCOELASTIC MODEL
Professor Castello (Federal University of Rio de Janeiro)
1 pm
303S-561

In this talk it will presented a strategy to determine the level of confidence of a model used to describe viscoelastic behavior. This strategy encompasses model calibration and the assessment of its predictive capabilities in different experimental scenarios.

31 May 2012
The Korteweg-de Vries Equation and Swirling Flow
Dr Steve Taylor (Auckland University)
12 pm
303-412

The Korteweg-de Vries (KdV) equation is one of the most famous nonlinear partial differential equations. It was first used to explain solitary waves in canals but has since found many other applications, including rich connections to scattering theory. In this talk I briefly describe Wang and Rusak's recent work in which the KdV equation shows up in a long-wave approximation for swirling flow of a fluid. Then I discuss how the KdV equation sheds light on novel ways to control swirling flow. I finish by showing how this new setting for the KdV equation sheds light on previously unknown and somewhat surprising properties of the equation itself.

24 May 2012
Calculus in the past with multiple delays arising in cancer cell modelling
Professor Graeme Wake (Massey University at Albany)
12 pm
303-412

Non-local calculus is often overlooked in the mathematics curriculum. We will outline an interesting new class of problems. Cells, especially cancer cells, move through the various phases whilst simultaneously undergoing growth and division. The biomass, taken here to be unstructured in size or position, moves through these stages following a multiple-stage delay system. The system is linear and taken to be autonomous. It is possible to reduce the solution of such a system to that of a nonlinear matrix eigenvalue problem which will be described in this paper. It will be illustrated with case studies including that in the paper that stimulated this work: by Simms K, Bean N , Koeber A. to appear in the Bull. Math. Biol., 2012.

17 May 2012
Fixed Point Theorems and Applications
Assoc. Prof. Warren Moors ( Auckland University)
12 pm
303-412

In this talk I will present some classical fixed point theorems and give some of their proofs. I will also give some applications of these fixed point theorems. Some of these applications will be known to you, but perhaps not their proofs.

10 May 2012
Waves in mathematical models of intracellular calcium and other excitable systems
Dr Wenjun Zhang (Auckland University)
12 pm
303-412

Excitable systems of reaction-diffusion equations are used to model many biophysical processes, including changes of intracellular calcium concentration in various cell types. Intracellular calcium plays an important role in many cells, being involved in the process of delivering external signals to the inside of the cell. Signaling is thought to occur in many cases via oscillation of the calcium concentration inside the cell. It is known that the dynamics of many mathematical models of intracellular calcium is strongly influenced by the presence of global bifurcations, including homoclinic and heteroclinic bifurcations of periodic orbits. Using a simple calcium model, we illustrate a numerical method, based on Lin's approach, for finding and continuing heteroclinic connections between periodic orbits. Locating such bifurcations helps to understand the overall bifurcation structure of calcium dynamics. This approach can also apply to other excitable models.

3 May 2012
Spike-adding mechanisms in transient bursts
Professor Osinga (Auckland University)
12 pm
303-412

Dynamical systems tools are designed to explain the asymptotic behaviour of a system, that is, what happens after transients have died out. In many applications, however, the transients play an important role and the asymptotic behaviour is of no interest. In this talk we explore how standard tools from dynamical systems can be used to analyse transient rather than asymptotic behaviour. To illustrate these ideas, we use the example of an excitable neuron model that is subject to a short current injection.

26 April 2012
Variable order and stepsize for numerical integrators
Professor John Butcher (Auckland University)
12 pm
412

For classical numerical methods for ordinary differential equations, there are inherent difficulties in constructing efficient and reliable variable order and variable stepsize algorithms. Denote by h the current stepsize and p the order of the current step. In the case of Runge-Kutta methods, for example, embedding is not able to provide asymtotically correct error estimates at a moderate cost. On the other hand, for linear multistep methods, the difficulties lie in the need to interpolate and reorganize the data when a variation of h or p is called for. Many of these difficulties can be overcome in the case of multistep-multivalue, or general linear, methods. The underlying idea is the use of a Lagrange multiplier formulation to provide a rational process for choosing among competing p and selecting the optimal h value. A second underlying idea is the use of the so-called `scale and modify technique to guarantee smooth transitions between one step and the next, even though h, and possibly also p, has been adjusted.

5 April 2012
Extrapolation of symmetrized Runge-Kutta methods
Annie Gorgery (Auckland University)
12 pm
412

Gragg introduced smoothing for the explicit two-step midpoint rule to dampen the parasitic oscillatory component arising naturally in the numerical solution. This allowed the development of successful extrapolation algorithms for nonstiff problems. The same smoothing formula could be used for solving stiff problems by the implicit midpoint or trapezoidal rules to dampen the stiff components. This has been successfully exploited by Lindberg, Dahlquist and others in developing extrapolation algorithms for stiff problems. For higher order methods, the generalization of smoothing is known as symmetrization. In this talk, we discuss two ways of applying symmetrization, active and passive, and four ways of subsequently applying extrapolation in the constant stepsize setting. We observe numerically that passive symmetrization with passive extrapolation is more efficient than active symmetrization with active extrapolation. In the variable stepsize setting, there are two ways of applying extrapolation with symmetrization. We also find that symmetrization can be used for error estimation. In the numerical experiments on the STIFF DETEST problem set, we observe that passive symmetrization with active extrapolation is more efficient than active symmetrization with active extrapolation. In summary, our work shows that extrapolation algorithms can be developed that are based on high order symmetric Runge-Kutta methods with appropriate use of symmetrization.

29 March 2012
Exponential decay for linear time-varying systems
Dr Adrian Hill (University of Bath, UK)
12 pm
Room 412

Consider the ODE dy/dt=A(t)y in R^N, and suppose that for each fixed t, solutions of dx/ds=A(t)x decay exponentially with s, at a uniform rate. Despite these hypotheses, y(t) may grow if A(t) varies too rapidly. We explain how this can happen, and give sufficient conditions for y(t) to decay exponentially. We also explore the related discrete case. (joint work with Achim Ilchmann, Ilmenau University, Germany)

22 March 2012
Canards in the dynamics of aircraft as ground vehicles
Professor Bernd Krauskopf (Auckland University)
12 pm
room 412

Aircraft are designed to fly, but are quite awkward vehicles on the ground that need to moved around the airport in a fast, reliable and safe fashion. The talk will discuss a particular limit to such safe operation via the loss of lateral stability while the aircraft is turning on the ground. In fact, we find a very rapid transition from small-amplitude oscillations to large-amplitude relaxation oscillations over a very small parameter range. This phenomenon, known as a canard explosion, is well known in the context of electrical, chemical and biological oscillators. In the context of a turning aircraft on the ground it is shown to be directly related to the successive saturation of tyre forces at the two main landing gears.

15 March 2012
Stochastic dynamics and the adaptive immune system
Dr Grant Lythe (University of Leeds)
12 pm
303-412

An important part of the body's immune response is the encounter between T cells and antigen-presenting cells inside lymph nodes, where their motion appears to be random. I will discuss ways to estimate timescales under the simplest hypothesis, that cell motion is Brownian, and numerical methods calculating collision rates between cells. The hypothesis that the apparent random motion exhibited by T cells in lymph nodes is due to motion on a preformed random spatial network has been explored in recent work with Graham Donovan. Everyone welcome!

8 March 2012
Global Invariant Manifolds at Orientable and Nonorientable Homoclinic Bifurcations
Dr Pablo Aquirre (Bristol University, UK)
12 pm
303-412

Bifurcations of vector fields explain crucial transitions between two (or more) different kinds of qualitative dynamics in systems. We are interested in the role of certain classes of homoclinic bifurcations as mechanisms by which global two-dimensional stable manifolds rearrange themselves both topologically and geometrically. At a codimension-one homoclinic bifurcation to a saddle with real eigenvalues, the stable manifold of the equilibrium is either an orientable or nonorientable surface. The change of orientation occurs at a class of codimension-two bifurcations, called inclination flip and orbit flip bifurcations, respectively. Indeed, at any such flip bifurcation, the stable manifold of the saddle equilibrium turns from a topological cylinder into a Mobius strip (or vice versa). Our approach is to study how this manifold rearranges itself globally near the codimension-one (non)orientable homoclinic bifurcations and also near the codimension-two inclination flip bifurcation, in the case that the unfolding results in a single stable periodic orbit. In this way, we are able to understand the overall organization of the dynamics in phase space and, in particular, how the change of orientability of the manifold affects the topology of the basin of attraction of the bifurcating limit cycle.

8 March 2012
Global Invariant Manifolds at Orientable and Nonorientable Homoclinic Bifurcations
Dr Pablo Aquirre (Bristol University, UK)
12 pm
To be announced

Bifurcations of vector fields explain crucial transitions between two (or more) different kinds of qualitative dynamics in systems. We are interested in the role of certain classes of homoclinic bifurcations as mechanisms by which global two-dimensional stable manifolds rearrange themselves both topologically and geometrically. At a codimension-one homoclinic bifurcation to a saddle with real eigenvalues, the stable manifold of the equilibrium is either an orientable or nonorientable surface. The change of orientation occurs at a class of codimension-two bifurcations, called inclination flip and orbit flip bifurcations, respectively. Indeed, at any such flip bifurcation, the stable manifold of the saddle equilibrium turns from a topological cylinder into a Mobius strip (or vice versa). Our approach is to study how this manifold rearranges itself globally near the codimension-one (non)orientable homoclinic bifurcations and also near the codimension-two inclination flip bifurcation, in the case that the unfolding results in a single stable periodic orbit. In this way, we are able to understand the overall organization of the dynamics in phase space and, in particular, how the change of orientability of the manifold affects the topology of the basin of attraction of the bifurcating limit cycle.

1 March 2012
Theory of vortex breakdown phenomenon
Shixiao Wang (Auckalnd University)
12 pm
301-254

`Vortex breakdown' is referred to the sudden and abrupt burst of a concentrated vortex flow. It is a widespread phenomenon that critically affects a variety of fluid flows of great importance. Although vortex breakdown has been the subject of studies for over five decades, no consensus has been yet reached as to the underlying physical mechanisms that govern it! The difficulty in understanding VB arises from the overwhelming nonlinear nature of NS equations. Basically, between 1957-1995, the studies have been limited to a `local analysis’’. Several competitive theories (T.B. Benjamin, S. Leibovich, H. Ludwieg) existed at the time but each of them only achieved a limited success. Wang & Rusak established the first consistent theory of VB by developing a novel `global analysis’ of the flow equations. The theory unified most of the known `local analyses’ and lead to the discovery of the instability mechanism that is essential for our understanding of VB. The talk will focus on the basic ideas rather than detailed mathematics. Some recent substantial progress along the line will be mentioned, if time permitted.

1 March 2012
Stochastic dynamics and the adaptive immune system
Dr Grant Lythe (University of Leeds)
12:00 pm
301-254

An important part of the body's immune response is the encounter between T cells and antigen-presenting cells inside lymph nodes, where their motion appears to be random. I will discuss ways to estimate timescales under the simplest hypothesis, that cell motion is Brownian, and numerical methods calculating collision rates between cells. The hypothesis that the apparent random motion exhibited by T cells in lymph nodes is due to motion on a preformed random spatial network has been explored in recent work with Graham Donovan.

22 February 2012
Mandelbrot Polynomials and Matrices
Professor Corless (University of Western Ontario)
2 pm
303S-279

In this talk, we explore a family of polynomials whose roots are related to the Mandelbrot set. These roots correspond to the k-periodic points of the iteration defining the Mandelbrot set. We discuss some of a variety of approaches to compute the roots of these polynomials; classical iterative schemes, eigenvalues of companion matrices and a novel family of recursively defined matrices. Time permitting, we look at an experimentally-discovered asymptotic series for the largest-magnitude roots. Joint work with Piers W. Lawrence and David J. Jeffrey.

19 January 2012
Noise and Diffusion in Models of Population Dynamics
Professor Horst Malchow (University Osnabrück, Germany)
2 pm
439-G10 (Symonds st.)

The dynamics of spatial and spatiotemporal pattern formation in nonlinear systems far from equilibrium are of continuous interest and many mechanisms of structure generation are not known yet. Here, the fascinating variety of spatiotemporal patterns in such systems and the governing mechanisms of their generation and further dynamics are described and related to plankton communities. The formation and spread of spatiotemporal structures in a simple predation-diffusion model with Holling type II or III predator is demonstrated. The analysis of the local system yields a number of stationary and/or oscillatory regimes. Correspondingly interesting is the spatiotemporal behaviour, modelled by reaction-diffusion equations. Spatial spread will be presented as well as competition of concentric and/or spiral population waves with non-oscillatory sub-populations for space, and long transients to spatially homogeneous population distributions. Environmental fluctuations are modelled as parametric as well as external multiplicative noise, using stochastic partial differential equations. The noise can enhance the survival of a population that would go extinct in a deterministic environment. In the parameter range of excitability and slow-fast dynamics of prey and predator, respectively, noise can induce local and global oscillations as well as local coherence resonance and global synchronization. Stationary patterns have been observed, too. Furthermore, it is shown that noise can suppress periodic travelling waves and the onset of chaos. The results are related to plankton dynamics, partly with viral infections of the prey population [1–4]. References [1] Malchow, H., Petrovskii, S. V. & Venturino, E. (2008). Spatiotemporal patterns in ecology and epidemiology: Theory, models, simulations . CRC Mathematical and Computational Biology Series, CRC Press, Boca Raton. [2] Sieber, M., Malchow, H. & Petrovskii, S. V. (2010). Noise-induced suppression of periodic travelling waves in oscillatory reaction-diffusion systems. Proceedings of the Royal Society A 466, 1903–1917. [3] Sieber, M., Malchow, H. (2010). Oscillations vs. chaotic waves: Attractor selection in bistable stochastic reaction-diffusion systems. The European Physical Journal – Special Topics 187, 95–99. [4] Petrovskii, S., Morozov, A., Malchow, H. & Sieber, M. (2010). Noise can prevent onset of chaos in spatiotemporal population dynamics. The European Physical Journal B 78, 253–264.