Evolution Equations: Theory, Numerics and Applications

**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 $(*)$.