Nikolay Kirov

Teaching
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INTRODUCTION IN

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CHAOTIC DYNAMICAL SYSTEMS

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(One-dimensional dynamics)

**One semester, 2 hours lectures + 2 hours labs weekly**
Lately, the notions dynamical systems and chaos became actual among
the specialists in various scientific fields. The aim of this course is
to introduce and to develop some fundamental ideas from the chaotic dynamics
using possibly simplest mathematical instruments. The contemporary computer
resources for modeling and visualization of dynamical systems allow their
efficient applications in educational process.

It turns out that the most of chaotic dynamics effects can be found
in the one-dimensional dynamics and even in a few simple samples. The logistics
mapping *F*(*m*)=*mx*(1-*x*) and the mapping shift
left, defined in the space of infinite sequences of two symbols (0 and
1) produce such dynamical systems which will be fundamental objects of
our studying.

The course is useful for students in both specialty - mathematics and
informatics and require background in Analysis - first part and ability
of using computers in elementary level. So, this course is available even
for the students of II and III courses.

Topics to be covered include:

Quadratic map
Hyperbolicity
Symbolic dynamics
Topological conjugancy
Chaos
Structural stability
Sarkovskii's theorem
The Schwarzian derivative
Bifurcation theory
Maps of the circle
Morse-Smale diffeomorphisms
Homoclinic points and bifurcations
The period doubling route to chaos
The kneading theory
Genealogy of periodic points

There will be a few labs consisting of examples discussion. Other
labs will be hold in the computer laboratory. Using a special software
system DYSY and MATHEMATICA package, the behaviour of some dynamical systems
we will be investigated.

The textbook for the course is the wonderful Devany's
book [1].

REFERENCES

1. Robert L. Devaney, **An Introduction to Chaotic Dynamical Systems**,
Addison-Wesley, 1989.

2. R. A. Holmgren, **A First Course in Discrete Dynamical Systems**,
Springer-Varlag, 1994.

3. L. S. Block, W. A. Coppel, **Dynamics in One Dimension**, (Lect.
Notes in Math. 1513), Springer, 1992.

4. Christian Mira, **Chaotic Dynamics**, World Scientific, 1987.

Last updated by
nkirov@math.bas.bg
on February 4, 1997