Applied Dynamical System Seminars (Winter 2007)

HH3017, Wednesday 4:00-5:00pm

1. Yu, Jin (Jan 24,2007), Memorial University of Newfoundland, Due to snow-storm, this was re-scheduled to Feb 7 at HH-3017.

Title: Spatial Dynamics of A Population Model With Dispersal

Abstract:

We study spatial dynamics of a class of integro-differential equations which describe the population dispersal process via a dispersal kernel. By appealing to the theory of asymptotic speeds of spread and traveling waves for monotone semiflows, we establish the existence of the spreading speed $c^{\ast}$ and the nonexistence of traveling wave solutions with the wave speed $c<c^{\ast}$. Then we use the method of upper and lower solutions to obtain the existence of monotone traveling waves with the wave speed $c\geq c^{\ast}$. It turns out that the spreading speed coincides with the minimal wave speed for monotone traveling waves.

2. Dr. David Iron (Dalhousie University), Jan 31, Wednesday at HH-3013.(Departmental Colloquium joint with Applied Dynamical System Seminar)

Title: Hot Spot Solutions to Microwave Heating Models

Abstract :

A class of ceramic materials have a temperature dependent conductivity. When such materials are heated in a microwave, the interaction between the electric field and the temperature of the sample results in a heat equation with a nonlocal source term. This system can give rise to solutions with highly localized regions of elevated temperature or hot-spots. We will present results for the stability and dynamics of such hot-spot solution for two models of temperature dependant conductivity. For both models, we will consider a long thin sample (1-D) and a thin plate-like sample (2-D).

3. Dr. James Watmough (University of New Brunswick)(Departmental Colloquium joint with Applied Dynamical System Seminar)

Time: 4:00-5:00pm Feb 12, Monday at SN--1019.

Title: The final size of an epidemic

Abstract: The early disease transmission model of Kermack and McKendrick established two main results that are still at the core of most disease transmission models today: the basic reproduction number,$\mathcal{R}_o$, as a threshold for disease spread in a population; and the final size of an epidemic. As models become more complex, the relationships between disease spread, final size and $\mathcal{R}_o$ are not as clear; yet $\mathcal{R}_o$ remains the main object of study when comparing control measures. In this talk I review the final size relation for a simple epidemic model and discuss its form in more complex models for treatment and control of influenza and HIV.

4. Ms. Rui HU (Memorial University of Newfoundland), Feb 28, Wednesday at HH-3017

title: Interaction between delta shock wave and nonlinear classical waves

Abstract:

A model of hyperbolic systems of conservation laws is considered. By

solving the initial value problem with three constants, the

interaction between $\delta-$shock wave and nonlinear classical

waves is studied. Under suitable generalized Rankine-Hugoniot

relation and entropy condition, five different structures of

interaction are established uniquely.

5.Dr. Yuan Yuan (Memorial University of Newfoundland)

Place: HH-3017 Time 4:00-5:00pm March 7th, Wednesday.

Title: Stability switches and Hopf bifurcations in a pair of delay coupled oscillators

Abstract In the talk, we consider a pair of delay-coupled limit cycle oscillators. Regarding the arithmetical average of two delays as a parameter, we investigate the effect of time delays on its dynamics. We show that there exist stability switches for time delays under certain conditions, which do not occur for the corresponding coupled system without time delays. We give detailed and specific conditions determining the amplitude death for different delays. On the other hand, we also investigate Hopf bifurcations induced by time delays using the normal form theory and center manifold reduction. In the region where the stability switches may occur, we can not only specifically determine the direction of Hopf bifurcations but also show the bifurcating periodic solutions are orbitally asymptotically stable. Numerical simulation results are also given to support the theoretical predictions.

6.Dr. Chun-hua Ou (Memorial University of Newfoundland)

4:00pm, Wednesday, March 14, 2007 HH-3017

Title:The Flame-Front Interface in Gas-Combustion: Modeling, Analysis and Computation

Abstract:

Metastable dynamics for a nonlocal PDE modeling the upwards propagation of a flame-front interface in a vertical channel is analyzed in the one-dimensional case where the channel cross-section is taken to be the slab -1<x<1. In a certain asymptotic limit, the interface assumes a roughly concave parabolic shape, and the tip of the parabola drifts asymptotically exponentially slowly towards the boundary of the domain. In contrast to previous analyses that studied this behavior by transforming the governing nonlocal PDE to a convection-diffusion equation, a novel nonlinear transformation is introduced that transforms the problem to a singularly perturbed quasilinear PDE. The steady-state problem for this transformed PDE, for which the parabolic interface shape maps onto a one-spike solution, is closely related to a class of two-point boundary value problems with seemingly spurious solutions studied initially by G. Carrier in 1968. Rigorous and formal asymptotic results for a one-spike solution to this transformed PDE are obtained together with a formal metastability analysis of certain time-dependent solutions.

7. Dr. Sergey Sadov Memorial University of Newfoundland

4:00-5:00pm, Wed, March 21, 2007

ˇ°How long is the tongue (or rather how narrow)ˇ±

Abstract:

Parametric resonance, also known as phase lock-in, is a phenomenon that prevents us from ever seeing the back side of the Moon. Another example is an upside-down pendulum: if the suspension point is subject to forced harmonic oscillations in the vertical direction, then surprisingly the pendulum may stay erect forever. I will describe two mathematical set-ups that exhibit parametric resonance: the spectral problem for a linear second order ODE with periodic coefficients (a Mathieu-type equation), and a nonlinear perturbation of a linear (Lissajous) flow on the 2D torus. Equations are different in the two cases, but they give rise to similar stability diagrams (``Arnold's tongues''). Asymptotics of tongue's width for small values of the perturbation parameter can be explained in terms of geometry of Fourier-supports of high-order perturbation series, which sheds light on the calculationsof Levy and Keller (1963), Hochstadt (1964), and Arnold (1959;1983). The analysis of Fourier supports reveals two different mechanisms of cancellation of terms up to a high order in perturbation theory: ``metric'' (for the spectral problem) versus ``arithmetic'' (for flows on a torus).

8. Dr. Andy Foster, Department of Mathematics and Statistics, Memorial University of Newfoundland

Place: HH3017 at 4:00-5:00pm 28 March, 2007, Wednesday Speaker:

TITLE: Bifurcations in a Dynamical Systems Model for Financial Asset Pricing

ABSTRACT: Dynamical systems modelling has been applied recently to the study of financial asset pricing. These dynamical systems models are constructed by considering the interacting strategies of various trader groups, and typically result in models that are nonlinear n-dimensional maps. In this talk, I will discuss the bifurcation behaviour of a relatively simple model of this type (Westerhoff, 2005). I will show that the model is properly written as a nonlinear planar map. Planar mays can generate much interesting behaviour, some analogous to scalar maps and some analogous to higher-dimensional ODE systems. These connections will be explored in some detail in this talk. The present model exhibits local period doubling and Neimark-Sacker bifurcations, which can be described analytically. It also displays "butterfly" homoclinic bifurcation and boundary crisis, which are global bifurcations and can only be studied by numerical methods.

Dr. Theodore Kolokolnikov(Dalhousie University), HH-3017 at 3:00pm-4:00 April 19, Thursday (Departmental Colloquium )

Title: Transition from stripe to spot patterns: the importance of the exponentially small.

Abstract:

Many one dimensional reaction-diffusion systems exhibit localized patterns that consist of back-to-back interfaces (box patterns) or spikes. When these patterns are trivially extended into a second dimension, one gets a stripe. Typically, such a stripe is unstable if its profile is a spike, but can be stable if its profile is a box. In this talk, we concentrate on the transition regime between the spike and box patterns. We find that the box-stripe is destabilized as the width of its profile is decreased. The key to this instability is the presence of exponentially small terms due to interface interaction. These terms grow as the width shrinks and become crucial when the width is logarithmically small.