Contributions individuelles
Mardi, 18 juin
Salle George V
16h00
The Sigma-Delta Modulator as a Chaotic Nonlinear Dynamical System
Campbell, Donald
University of Waterloo

Sigma-Delta modulators (or noise shapers as they are also called) are extensively used for analogue-to-digital and digital-to-analogue data conversion (signal processing).  Their dynamical behaviour can appear chaotic.  I will explore this behaviour from the point of view of nonlinear dynamical systems analysis.  To begin, the difference equation model of the sigma-delta modulator is introduced, and some basic results for bounded stability are obtained.  The model is cast formally as a discrete dynamical system, and important continuity results allowing for a linear analysis are established.  Drawing on this, I conduct a theoretical study of conditions for chaos or nonchaos using an adapted definition of Devaney's definition of chaos.  This study is extended to the dithered system, in the context of allowing stochastic aspects in the model.  I then introduce a stochastic formulation of the long-run dynamics, which is applied to give conditions for uniformly distributed error behaviour - conditions under which important consequences arise when dither is used to control the error statistics.

16h30
ANNULÉE
Spatial dynamics of flower organ formation
Cortés Poza, Yuririaa and Pablo Padilla Longoria
IIMAS, Universidad Nacional Autónoma de México

Understanding the emergence of biological structures and their changes is a complex problem. On a biochemical level, it is based on genetic regulatory networks (GRN) consisting on interactions of genes responsible for cell differentiation and coupled in a greater scale with external factors. As a specific example we consider the GRN involved in cellular determination during the early stages of development of the flower in  Arabidopsis thaliana plant.

We present a systematic way to derive evolution equations, based on experimental information, taken from the architecture of the flower’s GRN. Our model reproduces correctly the spatial dynamics of flower organ formation.

References

1. Alvarez-Buylla, E.R.; Chaos A.; Cortés-Poza Y.; Espinosa-Soto C.; Padilla-Longoria. De genes a patrones: una propuesta metodológica. Pp. 473-486.  In García- Colín, Scherer, L.; Dagdug, L.; Miramontes, P.; Roja, Arturo (Coord.). La Física Biológica en México: Temas Selectos. El Colegio Nacional. México. 2004.
2.  Alvarez-Buylla, E., Benítez M., Chaos, A.; Cortés-Poza, Y.; Esponosa-Soto, C.; Escalera, G., Padilla-Longoria. Floral Morphogenesis Stochastic Explorations of a Gene Network’s Epigenetic Landscape, Plos one, Vol. 3, Issue 11, noviembre 2008.
3.  Elena R. Alvarez-Buylla, Mariana Benítez, Álvaro Chaos, Yuriria Cortés, Gerardo Escalera-Santos, Carlos Espinosa, Pablo Padilla. Variational Problems Arising in Biology, Centre de Recherches Mathématiques, CRM Proceedings and Lecture Notes. Vol. 44, 2007.
4. Barrio, R. A., Romero-Arias, J. R., Noguez, M. A., Azpetia E., Ortiz-Gutierrez, E., Hernandez-Hernandez, V., Cortes-Poza, Y., Alvarez-Buylla, E., Cell patterns  emergence form coupled chemical and physical fields with cell proliferation dynamics: the Arabidopsis thaliana root as a study system, PLOS Computational Biology. Accepted, in printing, 2013.
5. Espinosa-Soto C, Padilla-Longoria P, Alvarez-Buylla E. R., A gene regulatory  network model for cell-fate determination during Arabidopsis thaliana flower development that is robust and recovers experimental gene expression profiles. Plant Cell 16: 2923–39, 2004.
6. Alvarez-Buylla ER, Benítez M, Dávila EB, Chaos A, Espinosa-Soto C, Padilla-Longoria P Gene regulatory network models for plant development. Curr Opin Plant Biol 10: 83–91. 2007.
17h00
Using Lyapunov Functions to Construct Lyapunov Functionals
McCluskey, Connell
Wilfrid Laurier University, Waterloo, Ontario

For many epidemic models written as

$$x (t) = f (x(t))$$
standard analysis includes calculating the basic reproduction number $\mathcal{R}_0$. Often, for $\mathcal{R}_0 > 1$ there is a globally asymptotically stable endemic equilibrium. In the last decade, this has been demonstrated for many ODE models through the use of Lyapunov functions based on the Volterra function

g(x) = x − 1 − \log x.
\label{eq1}

Other models include delay to better describe certain biological processes and can be written as $$x (t) = f (x(t), x(t − \tau))$$or, more generally,$$x (t) = f (x_t )$$where $x_t : [−\tau, 0] → {R}^n$ for some $\tau > 0$. Recently, Lyapunov functionals (based on the Volterra function $g$) have been used to resolve the global stability for many disease models with delay. In reviewing the many examples, it becomes clear that the Lyapunov functional for the delay equation is very strongly related to the Lyapunov function that works for the corresponding ODE.

In this work, we analyze the connection between the Lyapunov functional that works for Equation (4) and the Lyapunov function that works for the corresponding ODE. Using careful mathematical analysis, we obtain a test that allows one to classify, a priori, the terms that can incorporate delay without affecting the global asymptotic behaviour of the system.

17h30
Stability in a multiple-delayed neural network with distributed delay
Ncube, Israel
Mathematics Programme, Memorial University

We consider a network of three identical neurons incorporating distributed and discrete signal propagation time delays. The model for such a network is a coupled system of nonlinear delay differential equations. This presentation looks at the simple root single Hopf bifurcation of the trivial equilibrium of the network, and establishes some stability criteria. The results are based on the exact analysis of the characteristic equation of the system of delay differential equations.

18h00
Progress on Hard Thresholding Pursuit for sparse signal recovery //Progrès sur Hard Thresholding Pursuit pour la reconstruction de signaux parcimoniaux
Bouchot, Jean-Luc, Simon Foucart and Pawel Hitczenko
Drexel University

In this talk, theoretical results for the recovery of sparse vectors via the Hard Thresholding Pursuit algorithm are revisited. The main result states that all sparse vectors can be exactly recovered from incomplete linear measurements in a number of iterations at most proportional to the sparsity level as soon as the measurement matrix obeys a restricted isometry condition. Moreover we show that this recovery is robust against noise.

We also see that these results hold for a novel algorithm called Graded Hard Thresholding Pursuit which does not require a prior estimation of the sparsity level. We also show that under for certain particular shapes of vectors, a fixed sparse vector can be recovered with high probability in a number of iterations precisely equal to the sparsity level. These results are validated experimentally.

Dans cette présentation, les arguments théoriques pour la reconstruction de signaux parcimonieux par Hard Thresholding Pursuit sont revisités. Le résultat principal  établit en particulier que tout signal parcimonieux peut être reconstruit  à partir d’un nombre de mesures réduit en un nombre d’itérations au plus proportionnel  à  la sparsité des signaux considérés, dès lors que la matrice de mesure  a  une propriété d’isométrie réduite. On montre par ailleurs que cette reconstruction est robuste face au bruit.

On prouve  également que ces résultats sont valides pour une nouvelle variante de l’algorithme appelée Graded Hard Thresholding Pursuit pour laquelle une estimation a priori du niveau de parcimonie n’est pas nécessaire. Enfin on montre que pour certains types de vecteurs la  reconstruction d’un vecteur fixe est possible avec une grande certitude en un nombre d’itérations exactement  égal à  la sparsité du signal. Tous ces résultats sont également illustrés numériquement.