JV: Nonlinear Dynamics and Pattern Formation

News

2021 04 29      Problem set 3 is available as PDF file and in the wiki, where we have discussion pages for the exercises.

2021 04 22      Problem set 2 is available as PDF file and in the wiki, where we have discussion pages for the exercises.

2021 04 15      We agreed about the times for the seminars and the lectures as indicated below.

I updated problem 3 of problem set 1 to account for the fact that the centrifugal governor was not treated in class.
More details are given in the wiki, where I also provide discussion pages for the exercises.
Please contact me if you have are no account for the wiki.

2021 04 09      We will have our first lectures on 13 April 2021 at 15:00 in BBB (no access code required).

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".

13 April 2021 Pojman movie page» and Strogatz movie»

Problem Sets

Set    Date    Topic    PDF        Hints

01.     2021 04 09    Qualitative Analysis      (PDF file, 220kB) » ad 1.1d: How Grasshoppers Jump »

02.     2021 04 22    Basics in Dynamical Systems      (PDF file, 95kB) »

03.     2021 04 29    1D Maps      (PDF file, 90kB) »

Meetings

Course.      Tue    13 – 15

Wed 12 – 14



Exercise Sessions.              Mon    15 – 17

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.

Links

U. Behn      Literature List (PDF)      Links of Interest

P. Cvitanovic, et al      The Chaos Book

Scholarpedia      Dynamical Systems

2021 04 29	Problem set 3 is available as PDF file and in the wiki, where we have discussion pages for the exercises.
2021 04 22	Problem set 2 is available as PDF file and in the wiki, where we have discussion pages for the exercises.
2021 04 15	We agreed about the times for the seminars and the lectures as indicated below.
	I updated problem 3 of problem set 1 to account for the fact that the centrifugal governor was not treated in class. More details are given in the wiki, where I also provide discussion pages for the exercises. Please contact me if you have are no account for the wiki.
2021 04 09	We will have our first lectures on 13 April 2021 at 15:00 in BBB (no access code required).

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

Set	Date	Topic	PDF	Hints

01.	2021 04 09	Qualitative Analysis	(PDF file, 220kB) »	ad 1.1d: How Grasshoppers Jump »
02.	2021 04 22	Basics in Dynamical Systems	(PDF file, 95kB) »
03.	2021 04 29	1D Maps	(PDF file, 90kB) »

Course.	Tue	13 – 15
	Wed	12 – 14

Exercise Sessions.	Mon	15 – 17

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

SoSe2021. Pattern Formation and Nonlinear Dynamics