!DOCTYPE html> JV: Nonlinear Dynamics and Pattern Formation

SoSe2019. Pattern Formation and Nonlinear Dynamics

News

2019 06 06      •   Exercise sheet 14 is online.
2019 06 28      •   Exercise sheet 13 is online.
2019 06 21      •   Exercise sheet 12 is online.
2019 06 14      •   Exercise sheet 11 is online.
2019 06 11      •   I provided links to sound examples of frequency locking, and a nice desciption of the van der Pol oscillator in the Material section.
2019 06 09      •   We will have an interactive session starting from the metronomes that we saw in class, see Exercise sheet 10 for a pointer to the literature.
2019 06 01      •   Exercise sheet 9 is online.
2019 05 25      •   Exercise sheet 8 is online.
2019 05 19      •   Exercise sheet 7 is online.
2019 05 10      •   Exercise sheet 6 is online.
2019 05 05      •   Exercise sheet 5 is online.
2019 05 02      •   as further reading for exercises 1.1d there is a link now to the physics and physiology of Grasshoppers Jumps
     •   I provide Sage scripts for exercises 2.1, 3.1, and 3.2
     •   In section Material I provide Sage scripts for copweb plots and bifurcation diagrams that were discussed in the lecture
2019 04 19      •   The third exercise sheet is online
2019 04 12      •   The second exercise sheet is online
2019 04 05      •   The first exercise sheet is online
2019 04 03      •   In section Material I provide links to movies and literature discussed in the first lecture.
•   On request of a participant I added fractals to the list of Topics.
•   The exercise sessions will be on Tuesdays, 3-5pm, in SR114 at the ITP.
•   I corrected the announcement of the room for the lectures.

Topics

Dynamical Systems.      characterization of flows, critical points, stability
bifurcations, normal forms, center manifolds
chaos, routes to chaos, symbolic dynamics
 
Fractals.      fractal dimensions, multifractals, thermodynamic formalism
diffusion limited aggregation, snow flakes, breath figures
 
Pattern Formation. linear and non-linear stability
multiscale analysis, amplitude equations, Eckhaus and Benjamin-Feir instabilities
hydrodynamic systems, nonlinear waves, turbulence
reaction-diffusion systems, Turing instabilities

Background and Supporting Material for Lectures

I will use Python and Sage for numerical studies and visualization. The script files are notebooks that run interactively in Jupyter. Please store them on your computer, start Jupyter, and then open the notebook. The files contain absolutely minimalistic explanations and comments. I assume that you know what they are intended to do because we discussed the problems in class.

As introductions to Python and Sage I recommend the books Steward: "Python for Scientists" and Zimmermann u.a.: "Computational Mathematics with SageMath".

03 April 2019           Pojman movie page» and Strogatz movie»
research pages of Stefan Luther» and Katharina Krischer»
Hans Meinhardt: The Algorithmic Beauty of Sea Shells (Springer, 2009) and home page
Peter Goldreich, Sanjoy Mahajan, Sterl Phinney: Order-of-Magnitude Physics (draft for a book), in particular Chap. 1 and 2.3
23 April 2019           Sage scripts for cobweb plots and bifurcation diagrams (running in [sage -notebook jupyter]):
generalized shift map (420kB) »,     tent map (320kB) »,     logistic map (90kB) »
23 April 2019           Sage scripts for cobweb plots and bifurcation diagrams (running in [sage -notebook jupyter]):
generalized shift map (420kB) »,     tent map (320kB) »,     logistic map (90kB) »
6 June 2019           Listen to the injection locking sound example
and inspect the corresponding spectogram at the Injection locking page of Wikipedia.
11 June 2019           the dynamics of the van der Pol oscillator is nicely described in this scholarpadia article
the locking of metronomes is discussed by James Pantaleone, American Journal of Physics 70 (2002) 992.

Exercise Classes

Set    Date    Topic    PDF        Hints   
01.     2019 04 05    Qualitative Analysis      (PDF file, 800kB) » ad 1.1d: How Grasshoppers Jump »
02.     2019 04 11    Basics of Dyn Systems      (PDF-Datei, 88kB) » ad 2.1: Sage Skript for "Ball in Tube" (200kB) »
03.     2019 04 18    Conjugation and Symbolic Dynamics      (PDF-Datei, 88kB) » ad 3.1: Sage Skript for "Topological Conjugation" (160kB) »
            ad 3.2: Sage Skript for "Logistic Invariant Density" (120kB) »
04.     2019 04 25    Symbolic Dynamics      (PDF-Datei, 100kB) »
05.     2019 05 03    One-Dimensional Dynamics      (PDF-Datei, 90kB) »
06.     2019 05 10    Two-Dimensional Maps      (PDF-Datei, 100kB) »
07.     2019 05 17    Attractors, Saddles, and Fractal Sets      (PDF-Datei, 140kB) »
08.     2019 05 24    Hamiltonian Dynamics      (PDF-Datei, 90kB) »
09.     2019 05 31    Hamiltonian Chaos      (PDF-Datei, 95kB) » Sage Script: Driven Pendulum »
       Sage Script: Horse Shoe in the Driven Pendulum »
10.     2019 06 07    Wrapping up Nonlinear Dynamics      (PDF-Datei, 95kB) »
11.     2019 06 14    Linear Stability of Patterns      (PDF-Datei, 100kB) »
12.     2019 06 21    Linear Stability and Amplitude Equations      (PDF-Datei, 150kB) »
13.     2019 06 28    Pattern Competition and Hydrodynamic Instabilities      (PDF-Datei, 100kB) »
14.     2019 06 06    Water Waves      (PDF-Datei, 5MB) »
  

Meetings

Course.      Vollmer      Wed 11:15 – 12:45      ITP, Brüderstr    SR 114
          Thu 13:30 – 15:00      ITP, Brüderstr    SR 114
  
Exercise Sessions. Vollmer      Tue 15:15 – 16:45      ITP, Brüderstr    SR 114

Exams

There will be oral exams immediately after the end of the term.

Literature

G. Nicolis      Introduction to Nonlinear Science      Cambridge UP, 1995.     
H. Haken      Synergetics. An Introduction      Springer, 1983.     
E. Ott      Chaos in Dynamical Systems      Cambridge UP, 1993.     
M. Cross, H. Greenside      Pattern Formation and Dynamics in Nonequilibrium Systems      Cambridge UP, 2009.     
U. Behn      Literature List (PDF)      Links of Interest
P. Cvitanovic, et al      The Chaos Book     
Scholarpedia      Dynamical Systems