Nikolay Kirov



(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].

    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 on February 4, 1997