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URL: http://www.agnld.uni-potsdam.de/~shw/Lehre/lehrangebot/2006SS-DAM/2006SS-DAM.html

Klausur am 21. Juli (Freitag) von 11 bis 14 Uhr, Haus 19, Raum 422

Scheinabholung bei Frau Voigt (birgit AT agnld.uni-potsdam.de oder Tel. 1611) Haus 19, Raum 4.12, ab 14. August 2006.

43. Nonlinear data analysis and modeling in sciences

This lecture is one of the ``wahlobligatorischen Vorlesungen'' for the ``Wahlpflichtfach 1'' in ``Nichtlinearer Dynamik''.

Topics: Statistical tests, correlation function, spectra, stationarity, phase space reconstruction, recurrence plots, fractal dimensions, Lyapunov exponents, ODEs/PDEs, cellular automata, fractal models, networks, Markov models, ARMA processes, Langevin equations, Stochastic differential equations

This lecture is about the nature of mathematical modelling in physical sciences. The application of mathematical tools goes far beyond classical fields such as mechanics, thermodynamics, electrodynamics and quantum theory. Mathematical models nowadays also help to understand the flashing of fireflies, the relationship between heartbeats and respiration or else how the eyes move during reading. In this lecture we discuss how mathematical tools can be applied to answer questions like: How can we describe the flickering of a flame? What are the hidden rules behind the texture of an oil painting? How can we understand the highway traffic during the rush hour? How to model the sound of a violin? What do synchronously flashing fireflies have in common with an epileptic seizure? As the focus of the lectures is on the development of modelling techniques, we choose examples from various fields stressing applications which do not make part of ''classical physics''. We also will discuss, modelling in cognitive science and also models of neurons. We will introduce a wide range of modelling techniques, e.g. by ODEs, PDEs, autoregressive models, Langevin equations, nonparametric models, network models, neuronal networks, fractal models, and many more. Fitting the parameters of the respective models is an important issue, which will be treated in most of the cases. An important part of mathematical modelling is the analysis of the developed model. We will make use of computer simulations (C, perhaps parallel computing) and ready made programs such as IDL, Matlab and Maple. An important aspect will be the use of AUTO for bifurcation analyses. The lectures will complement lectures on nonlinear dynamics and time series analysis. It is thought to bridge between classical physics education and problems from neighbouring disciplines, preparing the participants for interdisciplinary research. Knowledge of mathematical methods is essential for the lecture. However, most of it can be found in Boas. Knowledge of a programming language, preferably C, is helpful but not necessary.

Course assessment: Solve exercises, test in the computer lab


Sommersemester 2006

J. Kurths & M. Thiel (V)
Do 15.15-16.45 1.09.1.15
Fr 11.00-12.30 1.08.0.50

M. Thiel & U. Schwarz (Exercises) (thiel or shw AT agnld.uni-potsdam.de)
Tue 11:15-12:45, Haus 19, Raum 423 (Computerpool)
Thu 11:15-12:45, Haus 19, Raum 423 (Computerpool)

Exercises:

DAM01.pdf, DAM02.pdf, DAM03.pdf, DAM04.pdf, DAM05.html,

Tools: matlab, TISEAN, gnuplot

Pfad fuer TISEAN-Binaries in der Bash-Shell setzen:
Die Datei .bashrc durch das Kommando

PATH=$PATH:/data/myscripts
export PATH
ergaenzen!
Gnuplot, Gnuplot (Aufruf: gnuplot):
plot  [1:12] sin (x) with line 5, exp(x)/2000
plot "< delay -d 1 henon.dat" w d
p [t=0:1] 3.5*t*(1-t), t
p 'name1' w l (i lp d st fst hist boxes) lt 2 lw 3, 'name2' 
Einige gnuplot Kommandos zur Gestaltung des Plots:
set size square
set xlabel 'X' set ylabel 'X'
set autoscale xy
set yrange [0.5:2.5]
set xrange [-20.:20]
set size
set xlabel ""
set ylabel ""
set nologscale y
set nologscale x

Rausschreiben einer PostScript-Datei unter gnuplot:

set term postscript landscape 'Helvetica' 14
set output 'name.ps' 
splot "<./delay -m3 -d10 w l name.dat"
set term x11
ode3solv.m, ode3.m.

Tipps zu TISEAN unter gnuplot:

Kovarianzfunktion:
plot"<./<corr -D200 AR2.dat"w lp

Powerspektren:
plot[0:0.1]"<./mem_spec -p40 -f400 AR2.dat"w lp
plot[0:0.1]"<./spectrum  AR2.dat"w lp

Phasenraum-Darstellung: 
plot "<./delay -m3 -d10 w l  name.dat" oder mit splot "<./delay -m3 -d10 w l name.dat"

Rekurrenz-Darstellung: 

plot "name.dat" w linesp
plot "<./recurr -d10 -r0.1 -%50 name.dat"

./recurr -d2 -r2 -%50 data.dat -o Darstellung per gnuplot plot "data.dat" w linesp und plot "data.dat.rec"

Mutual Information:
plot"<./mutual -b50 -D30 AR2.dat"w lp

Space-Time-Separation:
plot"<./stp -d2 -m2 AR2.dat"w lp

Bestimmung der Einbettungsdimension:
plot"<./false_nearest -m2 -M7 -d6  amplitude.dat" w l

Korrelations-Dimension:
d2 name.dat -d8 -t100 -o
plot 'name.dat.c2',.01*x**2.13

Lyapunov-Exponent:
lyap_k amplitude.dat -M6 -m3 -d8 -t100 -s500 -r.1 -o
plot[1:200]"amplitude.dat.lyap"w lp,-4.7+0.013*x

Lab mice:

henon -l10000 -o
Sun spots, Torus, x Lorenz, Noise, ElNino, AR 1, AR 2, Logistic map, SIN+Offset+Noise, harmonic process, Lorenz, Lorenz, Lorenz x, Lorenz y, Lorenz z,

Evaluation:

9 ECT-points all

Necessary condition to take part at the test: > 50 % of points of the exercises
Sufficient condition to get the ECT-points: > 50 % of points of the test

Literature:

J. Honerkamp, Stochastische Dynamische Systeme, VCH Weinheim 1990
R. Schlittgen & B.H. Streitberg, Zeitreihenanalyse, R. Oldenbourg, M"unchen 1997
Marek Fisz, Wahrscheinlichkeitsrechnung und mathematische Statistik
H. Kantz & T. Schreiber, Nonlinear Time Series Analysis, Camgridge University Press 1997 ** TISEAN 2.1:
http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.1/index.html
H. Rinne, Taschenbuch der Statistik, Harry Deutsch 2003
W.H. Press, S.A. Teukolsky, W.T. Vetterling & B.P. Flannery: Numerical Recipes in C, Cambridge University Press 1993
Script
An Introduction to Mathematical Modelling, E. Bender, Dover
Mathematical Modelling Techniques, T. Aris, Dover
The Nature of Mathematical Modeling, N. Gershenfeld, Cambridge
Mathematical Methods in the Physical Sciences, M. Boas, Wiley
Mathematisches Denken, T. Koerner
Gewoehnliche Differentialgleichungen, H. Heuser, Teubner
Mathematical Biology I/II, Murray
Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering.

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