## Mini-Symposium On Evolution Equations: Theory, Numerics and Applications

### Abstracts

Wolfgang Angerer, University of Vienna
Network of neurons coupled through diffusive neuromediators

A recent paper by Izhikevich et al. focuses on the role of neuromediators in memory and learning of neural networks. The resulting model is a scalar reaction-diffusion equation with global interaction, reflecting the philosophy of regarding a network as a collection of cells that interact only via the action of some transmitter (i.e., a neuromediator). We present the model and its behavior w.r.t. to associative memory and learning, and discuss possible extensions of the model to a system of ordinary differential equations coupled by PDE's.

Yongqiang Cao, York University
High Dimensional Data Clustering with Neural Networks

Most clustering algorithms do not work efficiently for data sets in high dimensional spaces. Due to the inherent sparsity of data points, it is not feasible to find interesting clusters in the original full space of all dimensions, but pruning off dimensions in advance, as most feature selection procedures do, may lead to significant loss of information and thus render the clustering results unreliable.

In a recent project with Jianhong Wu, we propose a new neural network architecture Projective Adaptive Resonance Theory (PART) in order to provide a solution to this feasibility-reliability dilemma in clustering data sets in high dimensional spaces. The architecture is based on the well known ART developed by Carpenter and Grossberg, and a major modification (selective output signaling) is provided in order to deal with the inherent sparsity in the full space of the data points from many data-mining applications. Unlike PROCLUS (proposed by Aggarwal et. al in 1999) and many other clustering algorithms, PART algorithm do not require the number of clusters as input parameter, and in fact, PART algorithm will find the number of clusters. Our simulations on high dimensional synthetic data show that PART algorithm, with a wide range of input parameters, enables us to find the correct number of clusters, the correct centers of the clusters and the sufficiently large subsets of dimensions where clusters are formed, so that we are able to fully reproduce the original input clusters after a reassignment procedure.

We will also show that PART algorithm is based on rigorous mathematical analysis of the dynamics for PART neural network model (a large scale system of differential equations with sigular perturbation), and that in some ideal situations which arise in many applications, PART does reproduce the original input cluster structures.

Jianhua Huang, Central China Normal University
Existence of traveling wave fronts for dealyed reaction diffusion equations with some zero diffusive coefficient

In this talk, I would like to talk about the existence of traveling wave fronts for delayed reaction diffusion equations with some zero-diffusive coefficient. Firstly, we tackle the existence of traveling wave fronts by monotone iteration method, and modify the operator in Wu & Zou [J. Dynm. Syst. Diff. Equs 2001], both monotonic nonlinear case and nonmonotonic nonlinear case are considered. Then we investigate the traveling wave fronts for delayed reaction diffusion equations with some zero diffusive coefficient without monotonicity by Schauder fixed point theorem. This work is joint with Prof. Xingfu Zou and Prof. Jianhong Wu.

Anatoli F. Ivanov,Pennsylvania State University
Periodic solutions for three-dimensional systems with time delays

We study the system

\begin{equation*}
(S)\left\{
\aligned  \dot y_1(t)  +  \mu_1 y_1(t)  & = f_1(y_2(t-\tau_2)) \\
\dot y_2(t)  +  \mu_2 y_2(t)  & = f_2(y_3(t-\tau_3)) \\
\dot y_3(t)  +  \mu_3 y_3(t)  & = f_3(y_1(t-\tau_1))
\endaligned \right.
\end{equation*}

of three differential equations with delays $\tau_j \geq 0$ and with decay coefficients $\mu_j>0, \; j = 1,2,3$,$, neurons coupled cyclically with delays. Our central assumption is that each$f_j$has either negative or positive feedback with respect to zero, and that the feedback is eventually negative. That is, for$ x \in \R\setminus\{0\}$and$ j = 1,2,3$one has \sign [f_j(x)\cdot x] = \sigma_j \in \{-1, +1\}, \text{ and } \sigma_1 \sigma_2\sigma_3 = -1. \leqno {\rm (H1)} (In particular,$f_j(0) =  0, \; j = 1,2,3$.)  Our main result is the following. a) There exist numbers$K_c >0$and$ K_u>0$(determined by$\mu_1, \mu_2, \mu_3$) such that the linearization of (S) at zero [i] has no real characteristic values if and only if$K > K_c$; [ii] has characteristic values with positive real part if and only if$K>K_u$. b) If$K > \max\{K_u, K_c\}$then system (S) has a nontrivial periodic solution. Yuriy Kazmerchuk, York University Theory, Stochastic Stability and Applications of Stochastic Delay Differential Equations: a Survey of Recent Results This survey summarizes recent results on stochastic delay differential equations (SDDE) updating the paper written by A.V. Swishchuk and A.F. Ivanov [1]. Most of the reviewed articles were published after the year of 1997. The problems discussed in this survey are Markov property of solutions of SDDE's, stochastic stability, elements of ergodic theory, numerical approximation, parameter estimation, applications in biology and finance. Dong Liang, York University Numerical Travelling Wavefronts in a Reaction Advection Diffusion Model with Nonlocal Delayed Effects We consider the growth dynamics of a single species population with two age classes and a fixed maturation period living in a spatial transport field. A Reaction Advection Diffusion Model with time delay and nonlocal effect is derived if the mature death and diffusion rates are age independent. We discuss the existence of travelling wave fronts for the cases when the birth functions are the ones appeared in the well-known Nicholson's blowflies equation, and we consider and compare numerical solutions of the travelling wave fronts for the problems with nonlocal temporally delayed effects and local temporally delayed effects. This is joint work with J. Wu. Shiwang Ma, Shanghai Jiaotong University Existence and stability of traveling wavefronts in a single species diffusion model with age structure In this paper, we study the existence, uniqueness and global asymptotic stability of traveling wave fronts in a non-local reaction-diffusion model for a single species population with two age classes and a fixed maturation period living in spatially unbounded environment. Under some weak conditions on the birth function, by the elementary super and sub solution comparison and the squeezing technique, we prove that the equation has exactly one non-decreasing traveling front (up to translation) which is monotonic and globally asymptotic stability with phase shift. This is joint work with J. Wu. Israel Ncube, York University Thoughts on stationary densities of some random algorithms We consider a certain class of discrete random neural network-type learning algorithms. Focussing on a suitable metric space of distributions, we are interested in the issue of existence of stationary densities for these algorithms. The presentation is an outline of some ongoing work in this direction. Minh Van Nguyen, Hanoi University of Science Evolution Semigroups and Almost Periodic Solutions of Evolutions Equations In this talk I will speak of a new method of study and some new results concerned with the existence of bounded and almost periodic mild solutions of inhomogeneous linear equations of the form u'(t)= A(t)u(t)+f(t), t\in {\mathbb R},  where$A(t)$are (unbounded) linear operators for$t\in {\mathbb R}$,$f$is a continuous and bounded on$R$. This problem is not only of interest in itself, but also of great importance in the qualitative theory of evolution equations such as the theory of invariant manifolds using the Lyapunov -Perron method. The method is based on the spectral analysis of the so-called evolution semigroups on function spaces and the harmonic analysis of bounded mild solutions to the inhomogeneous equations. Various spectral conditions for the existence and uniqueness of bounded and almost periodic solutions are found. Anatoliy Swishchuk, York University Evolutions of Population Biological Systems in Random Environment: Limit Theorems and Stability The lecture is devoted to the Limit Theorems and Stability of stochastic evolution equations arising in population biological systems in random environment. We consider the following biological systems in random environment: 1) an epidemic model; 2) genetic selection model; 3) demographic model; 4) Bellman-Harris branching model; 5) predator-prey model and 6) logistic growth model. We study averaging, merging, diffusion approximation and normal eviations of the evolution equations in series scheme, which describe these biological systems. We also state the stability results for the evolution equations in series scheme of such biosystems using the stability of averaged, diffusion and normal deviated evolution equations. Sergei Trofimchuk, University of Chile Global stability and chaos in a family of scalar functional differential equations We consider scalar delay differential equations$x'(t) = - \delta x(t) + f(t,x_t) \ (*)$with nonlinear$f$satisfying a sort of negative feedback condition combined with a boundedness condition. The well known Mackey-Glass type equations, equations satisfying the Yorke condition, equations with maxima are kept within our considerations. We analyze different types of behavior in$(*)$, presenting some results about the global asymptotical stability, the existence of multiple periodic solutions, and the existence of chaotic solutions in$(*)\$.