July 20, 2024

Southern Ontario Dynamics Day Workshop
University of Guelph & The Fields Institute,

April 7 2006

Organizers: Prof. Monica-Gabriela Cojocaru & Prof. Anna T. Lawniczak
Department of Mathematics & Staistics
Univeristy of Guelph

Supported by :

Department of Mathematics & Statistics,


Patrizia Daniele, University of Catania, IT:
Dynamic Networks and Evolutionary Variational Inequalities

We outline the importance of the evolutionary variational inequalities theory from the beginning in 1967 to the numerous applications in the field of operational research, engineering, economy, finance. The study of the evolutionary variational inequalities has been given a substantial boost in the last decade when it was proved that dynamic economic, financial and transportation equilibrium problems on networks can be expressed by means of evolutionary variational inequalities. So we have the possibility to follow the evolution in time of economic phenomena, since the theory of evolutionary variational inequalities allows for the study of existence, uniqueness and sensitivity analysis of the equilibrium trajectory. Also the problem of the calculation of the solution in a time interval can be solved by means of appropriate computational techniques and we present two computational methods based on the discretization procedure. The first one requires no regularity conditions on the solution and makes use of the mean value operators. The second one requires the continuity of the solutions and exploit the relation between the solutions to an evolutionary variational inequality and the stationary points of a projected dynamical system. Some numerical examples of traffic networks and spatial price models will be presented too.

Matt Davison, University of Western Ontario:
Probability Theory, Linear Operators, and Discrete Logistic Maps

It is well known (to those that know it well) that nonlinear dynamical systems acting on points have corresponding to them linear operators acting on functions. Systems of differential equations have corresponding to them partial differential operators, and when these systems are stochastic, the resulting partial differential operators lead to famous financial math results such as the Black -Scholes equation. The topic of this talk will be discrete dynamical systems, to which correspond the so-called Perron-Frobenius Operator. After giving a quick review of this theory I will use it to provide some results on weakly coupled logistic map systems.

Henryk Fuks, Brock University:
Dynamics of the language acquisition process

We will demonstrate how the paradigm of complex networks can be used to model some aspects of the process of acquisition of the second language. When learning a new language, knowledge of 3000-4000 of the most frequent words appears to be a significant threshold, necessary to transfer reading skills from the first to the second language. This threshold corresponds to the transition from Zipf's law to a non-Zipfian regime in the rank-frequency plot of words of the English language. Using a large dictionary, one can construct a graph G representing this dictionary, and study topological properties of subgraphs of G generated by the $k$ most frequent words of the language. Since the vocabulary grows with time, one can also think of this as a dynamical process in which $k$ increases as a function of time. The clustering coefficient of subgraphs of G reaches a minimum in the same place as the crossover point in the rank-frequency plot. We conjecture that the coincidence of all these thresholds may indicate a change in the language structure, which occurs when the vocabulary size reaches about 3000-4000 words.

Martin Golubitsky, University of Houston, USA & University of Toronto:
Patterns of Synchrony in Lattice Dynamical Systems

Lattice differential equations consist of an infinite number of coupled ODEs parametrized by a planar lattice (square and hexagonal being the most popular) such as would be obtained by discretizing a planar system of PDEs. We assume the system is homogeneous; that is, the ODEs at each site are identical.
A pattern of synchrony is a finite dimensional flow invariant subspace for all such lattice differential equations. Patterns of synchrony can be given by fixed-point subspaces of symmetry subgroups. We discuss how the existence of these patterns depends on the kind of coupling (nearest neighbor or nearest and next nearest neighbor). In joint work with Yunjiao Wang, Ana Dias, and Fernando Antoneli, we show that if there is enough coupling then all such patterns are spatially periodic. This statement is false for the nearest neighbor case.

Herb Kunze, University of Guelph
Fractal Tops & Colour Stealing

A hyperbolic iterated function system (IFS) has a unique fixed point that we refer to as its set attractor. To each point on the set attractor we can associate a code that tells us the order in which we should apply the IFS maps in order to reach or approach the point. By traveling through code space and using the notion of fractal tops (a very recent idea introduced by M. Barnsley), we connect the dynamics of two different IFSs: we paint the points of (an approximation of) the attractor of the first IFS by stealing colours from a digital image via the dynamics of the second hyperbolic IFS. Besides developing assorted theory, I will explore various examples and applications, producing some stunning images along the way.

Greg Lewis, University of Ontario Institute of Technology:
Bifurcations in a Differentially Heated Rotating Spherical Shell

Mathematical models of fluid systems that isolate the effects of differential heating and rotation are useful tools for studying the behaviour of large-scale geophysical fluids, such as the Earth's atmosphere. In this talk, I discuss the bifurcations of steady axisymmetric solutions that occur in one such model that considers a fluid contained in a rotating spherical shell. A differential heating of the fluid is imposed between the pole and equator of the shell, and gravity is assumed follow axisymmetric solutions through parameter space, and numerical approximations of normal form coefficients are computed for a cusp bifurcation that acts as an organizing centre for the observed dynamics.

Pietro Lio, University of Cambridge, UK:
Mathematical Model of HIV Superinfection Dynamics and R5 to X4 Switch

During the HIV infection several quasispecies of the virus arise, which are able to use different coreceptors, in particular the CCR5 and CXCR4 coreceptors (R5 and X4 phenotypes, respectively). The switch in coreceptor usage has been correlated with a faster progression of the disease to the AIDS phase. As several pharmaceutical companies are starting large phase III trials for R5 and X4 drugs, models are needed to predict the co-evolutionary and competitive dynamics of virus strains. We present a model of HIV early infection which describes the dynamics of R5 quasispecies and a model of HIV late infection which describes the R5 to X4 switch. We report the following findings: after superinfection or coinfection, quasispecies dynamics has time scales of several months and becomes even slower at low number of CD4+ T cells. The curve of CD4+ T cells decreases, during AIDS late stage, and can be described taking into account the X4 related Tumor Necrosis Factor dynamics. Phylogenetic inference of chemokine receptors suggests that virus mutational pathway may generate R5 variants able to interact with chemokine receptors different from CXCR4. This may explain the massive signalling disruptions in the immune system observed during AIDS late stages and may have relevance for vaccination and therapy.

Xinzhi Liu, University of Waterloo
Synchronization of Chaos and Impulsive Dynamical Systems with Time Delay

With the rapid development of personal communications and the Internet, information security has become an increasingly important issue of the telecommunication industry. Recently, there has been tremendous worldwide interest in exploiting chaos for secure communications. The idea is to use chaotic systems as transmitters and receivers, where the message signal is added to a chaotic carrier generated by the transmitter system and it is recovered at the receiver through synchronization. This talk will discuss the method of impulsive synchronization for chaos-based secure communication systems in the presence of transmission delay and sampling delay and the related theory of impulsive dynamical systems with time delay, which provides the main framework for modeling the error dynamics between the driving and response systems employed in such communication systems.

Marcus Pivato, Trent University:
Crystallographic Defects in Cellular Automata

Let L:= Z^D be the D-dimensional lattice, and let A^L be the Cantor space of L-indexed configurations in some finite alphabet A, with the natural L-action by shifts. A `cellular automaton' is a continuous, shift-commuting self-map F of A^L, and an `F-invariant subshift' is a closed, F-invariant and shift-invariant subset X of A^L. Suppose x is a configuration in A^L that is X-admissible everywhere except for some small region we call a `defect'. It has been empirically observed that such defects persist under iteration of F, and often propagate like `particles' which coalesce or annihilate on contact. We construct algebraic invariants for these defects, which explain their persistence under F, and partly explain the outcomes of their collisions. Some invariants are based on the cocycles of multidimensional subshifts; others arise from the higher-dimensional (co)homology/homotopy groups for subshifts, obtained by generalizing the Conway-Lagarias tiling groups and the Geller-Propp fundamental group.

Mary Pugh, Univerity of Toronto
A Finite Locus Effect Diffusion Model for the Evolution of a Quantitative Trait

If ten genes affect an individual's height and one has a population with average height 6'5" that is forced to live in low-ceilinged caves, how might this selective pressure act at the genetic level of individuals? What happens if the different genes have different magnitudes of effect? A nonlocal diffusion model is constructed and studied for the joint distribution of absolute gene effect sizes and allele frequencies for genes contributing to an quantitative trait in a haploid population where there is a selection pressure on the quantitative trait.
The model is designed to approximate a discrete model exactly in the limit as both population size and the number of genes affecting the trait tend to infinity. For the case where loci can take on either of two distinct effect sizes, not necessarily with equal probability, numerical solutions of the system indicate that response to selection of a quantitative trait is insensitive to the variability of the distribution of effect sizes when mutation is negligible.
This is joint work with Judy Miller and Matt Hamilton of Georgetown University.

Allan Willms, University of Guelph:
A Dynamic System for Real-Time Robot Path Planning

We present a simple yet efficient dynamic system for real-time collision-free robot path planning applicable to situations where targets and barriers are permitted to move. The algorithm requires no prior knowledge of target or barrier movements. In the static situation, where both targets and barriers do not move, our algorithm is a dynamic programming solution to the shortest path problem, but restricted by lack of global knowledge. In this case the dynamic system converges in a small number of iterations to a state where the minimal distance to a target is recorded at each grid point, and our robot path-planning algorithm can be made to always choose an optimal path. In the case that barriers are stationary but targets can move, the algorithm always results in the robot catching the target provided it moves at greater speed than the target, and the dynamic system update frequency is sufficiently large. We also look at how the algorithm can be modified to choose paths that not only reach the target via the shortest possible route, but also shun obstacles. The effectiveness of the algorithm is demonstrated through a number of simulations.

Gail Wolkowicz, McMaster University:
An Alternative Formulation for a Delayed Logistic Equation

After pointing out the problems with the classical delayed logistic equation as a model of population growth, an alternative expression for a delayed logistic equation is derived. It is assumed that the rate of change of the population depends on three components: growth, death, and intraspecific competition, with the delay in the growth component. In our formulation, we incorporate the delay in the growth term in a manner consistent with the rate of instantaneous decline in the population given by the model. After a complete global analysis of the model, the dynamics are compared with the dynamics of the classical logistic delay differential equation model, the classical logistic ordinary differential equations growth model, and various other more recent formulations. Implications of our analysis for including delay in such models is also discussed. This is joint work with Julien Arino and Lin Wang

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