Classifications of
Dulac maps and epsilon-neighborhoods
:
The project is a fundamental research project in mathematics aimed at applying
mathematical analysis tools in understanding the qualitative behaviour of
dynamical systems. The dynamical systems are mathematical models given as
systems of differential equations whose solutions describe natural processes.
Our motivation comes from bifurcation theory of dynamical systems. The most
important question is the question of stability with respect to the parameters:
can one predict how small changes in parameters will influence the long-term
behaviour of solutions. Qualitative theory is concerned with describing the
evolution of solutions in time without explicitly solving the equations. In
particular, to understand bifurcations, one can measure the complexity of
attractors of trajectories by the number of closed orbits they bifurcate into by
parameter changes. Even in very specific and seemingly simple models in the
plane, this question is open (the famous open Hilbert’s 16th problem).
One-dimensional representation of a system close to an attractor is given by
the function called the Poincaré map. The question of understanding
bifurcations of systems is translated into question of understanding intrinsic
properties of families of Poincaré maps. We study a special type of this
function, the so-called Dulac map, which corresponds to attractors of saddle
polycycle type. To understand this function more deeply, with collaborators
from geometry and dynamical systems group at the Institut de Mathématiques de
Bourgogne, Dijon, we would like to find a simple form to which it can be
translated, at the same time requesting that the translation preserves
properties of the original function. That is, as our final goal, we would like
to describe the analytic class of Dulac maps, as one step toward understanding
bifurcations of saddle polycycles. Dynamical systems are currently an important
and quickly developing branch of mathematics in the world, with non-neglectable
applications to technology. Given the fact that in Croatia there is only a
small group of people working in the field, one aim of this project is to
contribute to its propagation in Croatia and to its visibility among students
and future scientists by three means: 1. Strengthening the leader and the whole
group by means of new collaborations, through the proposed mobility program, 2.
Organizing a Dynamical systems workshop and inviting foreign experts to give
courses aimed at postgraduate / doctoral students and young researchers, 3.
Equipping the Central mathematical library with modern books and textbooks in
the field.
The main activities planned in the project are: the long-term stay of the
leader at University of Burgundy for scientific collaboration (6 months in the
first halfth of the project) and organization of the Workshop and conference on
dynamical systems in Zagreb in the second halfth of the project.