Seminar: Chaotic Dynamics

PD Dr. Pavel Gurevich

Dr. Nikita Begun

Schedule, Summer 2016

Seminar:

Monday 10.00-12.00, SR 130 / Arnimallee 3

Description

We will deal with chaotic phenomena in the theory of dynamical systems. The phenomenon of chaos was observed by Poincaré back in 19th century, but the rigorous mathematical theory has not been created until the second half of the 20th century and is still an actively developing field. In the seminar, we will understand what “chaos” means and how it may occur. We will concentrate on one- and two-dimensional diffeomorphism, but the ideas extend to much more general systems.

The language of the seminar is supposed to be English (with the help of German if needed).

We expect that the participating students have attended Analysis 1 and 2. Preliminary knowledge of the dynamical systems theory is desirable but not necessary. As a prerequisite for everybody, we strongly recommend to read Sections 1.1-1.4 (with exercises) in [Devaney, 1986].

Topics

One-dimensional maps

1. Quadratic family (Devaney 1.5)

2. Symbolic dynamics (Devaney 1.6) and Topological conjugacy (Devaney 1.7)

3. Introduction to chaos (Devaney 1.8) and Structural stability (Devaney 1.9).

4. Sarkovskii’s theorem: “period 3 implies chaos” (Devaney 1.10).

5. Bifurcations of diffeomorphisms (Devaney 1.12).

6. Maps of circles (Devaney 1.14) and Morse-Smale diffeomorphisms (Devaney 1.15).

7. Homoclinic points and bifurcations (Devaney 1.16).

8. Period-doubling route to chaos Devaney (Devaney 1.17-19, also another book is needed).

Two-dimensional maps

9. Smale Horseshoe (Devaney 2.3, Wiggins 23).

10. Hyperbolic toral automorphisms (Devaney 2.4).

11. Solenoid (Plykin attractor) (Devaney 2.5).

12. Lorenz and Roessler attractors - I (Hirsch-Smale-Devaney 14).

13. Lorenz and Roessler attractors - II (Hirsch-Smale-Devaney 14).

Literature

• Robert Devaney, An introduction to chaotic dynamical systems, 1986.
• Stephen Wiggins, Introduction to applied nonlinear dynamical systems and chaos, 2003.
• CMorris W. Hirsch, Stephen Smale, Robert Devaney, Differential equations, Dynamical systems and introduction to chaos, 2004.