Chapter 1 - Overture

C.N. Yang interview

Kerson Huang (Hong Kong University, July 29, 2000) [password needed]
A very personal and in parts hilarious overview of the 20th century physics - should you really be reading this book? Click here for few quotes.

A study of the Rössler system

Radford Mitchell, Jr. (spring 2005)

An exploration of the Rössler attractor

Gabor Simon (12 jan 2000)

Periodic orbit theory: A study of the Rössler attractor

Joachim Mathiessen (20 jan 2000)

Construction of Poincaré return maps for Rössler flow

Arindam Basu (summer 2007)

Stroboscopic map for a driven pendulum

Slaven Peles (2004)
y'' + y'/Q + sin(y) = r cos(at),
  code

3-d billiard Jacobians

Andreas Wirzba (2 Mar 1995)

Integrating helium dynamics

A. Prügel-Bennett
  mathematica code

AFM trajectories

Siddhartha Kasivajhula
Java applet which presents a stroboscopic section of a tapping mode Atomic Force Microscope, alongside with the (x,y) space trajectories. The program simulates trajectoris for given initial conditions and system paramaters.

Physicist's Pinball

William Benfold Java applet which presents a Poincaré section of the system, alongside the three discs. The program finds cycles with a desired itinerary and computes the escape rates.

A simple pinball simulator

A. Prügel-Bennett (Adam: Benfold's program is superior to this one)
  c code

Xpinball

A. Prügel-Bennett that allows you to view orbits and the Poincaré section in the 3-disk billiard. Requires Unix with X11 windows and Motif library
  code

GUI matlab billiard simulator

Steven Lansel and Mason A. Porter (3 May 2004, tested on releases 12 and 13)
a sample Poincaré section

Symbolic dynamics in chaotic systems

Kai T. Hansen (Ph.D. thesis 1993)

Generalized Markov coarse graining and the observables of chaos

Donal MacKernan (Ph.D. thesis 1997)

Symbolic Dynamics and Markov Partitions for the Stadium Billiard

Kai T. Hansen and Predrag Cvitanović (draft, 13 April 95)
(still a preprint:) An imperfect attempt to exemplify the nontrivial aspects of Markov diagrams, symbol planes, role of symmetry in context of a popular dynamical systems problem

Markov partition for collinear helium

K. Richter, G. Tanner and D. Wintgen
from "Classical mechanics of two-electron atoms", Phys. Rev. A 48, 4182-4196 (1993)

Soft Bunimovich Stadium

Sune Hørlück (8 sep 1995)
Soft (but not easy) Bunimovich stadium: A small investigation
  Kai's instructions for soft Bunimovich stadium (in Norwegian, 30 may 1995)

Periodic orbits: how to get them

Cristel Chandre (lecture notes, Sept 2001)

Periodic orbits of a forced pendulum

Cristel Chandre (Aug 18 2002)
An implementation of sect. "Newton method for flows", for a 2-degree of freedom Hamiltonian flow. Should be easily adoptable to other 2-degree of freedom Hamiltonian systems, such as the collinear helium.
  c code tarball

Construction of Poincaré return maps for Rössler flow

Arindam Basu (summer 2007)

Multishooting Mathematica code for periodic orbits of Rössler flow

Jon Newman (fall 2008)

A topologically guided method to find orbits in chaotic systems

Kai T. Hansen
Phys. Rev. E 52, 2388 (1995)

Refining periodic orbits

Carl P. Dettmann (April 2002)
Implementation of a search for periodic orbits of a flow within a Poincaré section

Finding billiard periodic orbits by line minimization

A. Prügel-Bennett
For the overview, see the solutions of chapter on fixed points exercises
  c code

Finding billiard periodic orbits by line minimization

Igor Romanovsky
A. Prügel-Bennett's routine for finding billiard periodic orbits by line minimization
  Microsoft Fortran90 code

Finding simple colinear helium periodic orbits

A. Prügel-Bennett
  helium_po.m contains various functions
  periodic_orbits.m illustrates how these are used to find periodic orbits

Cycle-finding programs

F. Christiansen (29 oct 96)
Preliminary version, mostly maps
  numerical routines package

Systematic detection of unstable periodic orbits in discrete chaotic dynamical systems

F.K. Diakonos, D. Pingel and P. Schmelcher (4 July 2000)

Routines for finding periodic orbits

Vakhtang Putkaradze (eternalized "preliminary version," 29 apr 1996)
Muddled instructions for using Putkaradze-Christiansen numerical routines.
  Cycle-finding programs for flows, F. Christiansen's and V. Putkaradze's programs for for finding periodic orbits and zeros of Fredholm determinants.
You will probably also need the sample data sets

Prime orbits and prime numbers

R. Mainieri
A quick overview of the parallels between prime numbers and prime orbits

Spectrum of the Liouville operator

Niels Søndergaard (30 aug 1995)

Periodic orbit theory of linear response, a sketch

Predrag Cvitanović (18 may 1998)

Dynamical zeta functions

A. Prügel-Bennett
Mathematica programs to construct the dynamical zeta function and Fredholm determinant
  orbits.m,   zeta.m,   fredholm.m

A logistic map repeller

P. Andrésen
The dynamical zeta function and Fredholm determinant for a logistic map repeller - solution of the chapter on cycle expansion exercise

Periodic orbit theory: A study of the Rössler attractor

Joachim Mathiessen (20 jan 2000)

Chaotic Radial Oscillations of a Harmonically Forced Gas Bubble, Parametric Dependence and Consequences for Sonoluminescence

Gabor Simon (2 feb 2000)
"Periodic orbit theory applied to a chaotically oscillating gas bubble in water"
(with G. Simon, M.T. Levinsen, I. Csabai, Á. Horváth and P. Cvitanović), Nonlinearity 15, 25 (2002)

Nonlinear dynamics of dispersion managed breathers in Gaussian Ansatz approximation

Rytis Paškauskas (2 feb 2000)

Comparison between cycle expansion and adjoint equations

Juri Rolf (11 Feb 1997)
In this project J. Rolf proposed new conjectures for an infinite family of nontrivial spectral determinants. The results were Rolf's contribution to ``Beyond periodic orbit theory'' of P. Cvitanović, G. Vattay, J. Rolf and Kim Hansen, Nonlinearity 11, 1209 (1998).

Why does the leading eigenvalue give escape rate?

Mario Sempf (April 2001)
Why, again?

Deterministic diffusion, sawtooth

Peter Andresén (3 Feb 1999)
Termpaper

Deterministic diffusion, sawtooth

Christian I. Mikkelsen (12 Jun 1999)
Termpaper

Deterministic diffusion, sawtooth

Khaled A Mahdi (22 Mar 1998)
  mathematica notebook termpaper

Deterministic diffusion, zig-zag map

Jakob Kisbye Dreyer (3 Jun 1999)
Termpaper

Hard Bunimovich stadium, washbord diffusion

Jonas Lundbek Hansen (23 aug 1995)
Termpaper

Introduction to chaos and diffusion

G. Boffetta, G. Lacorata and A. Vulpiani
nlin.CD/0411023

Fourth-order time-stepping for stiff PDEs

L N Trefethen (July 2002)
(published in SIAM J. Sci. Comp.)
  1-page, 1-second matlab ETDRK4 code for Kuramoto-Sivashinsky equation

Dynamical systems approach to 1-d spatiotemporal chaos - A cyclist's view

Yueheng Lan (Ph.D. thesis, Georgia Tech 2004)

Kuramoto-Sivashinsky simulations

Ruslan L. Davidchack (April 2007)
  A demo of the matlab code + other source files - improvements/additions are welcome

Analysis and numerical experimentation, Kuramoto-Sivashinsky system

1-week project (April 2007)

Kuramoto-Sivashinsky: 1. A fishing expedition; 2. Flickering flame front

Report by spring 2007 GaTech chaos class. "Temporary" forever: some results not yet included (April 2007)

Flame front: the movie

Kirill Davydychev (April 2007)
description
  [avi format]

Kuramoto-Sivashinsky weak turbulence

Evangelos Siminos (12 dec 2004)

Turbulence, and what to do about it?

1-week project (June 1999):
Involves analysis of a dynamical system (fixed points, stability, bifurcations) and numerical experimentation with integration of a set of differential equations describing the system.

Hopf's last hope: spatiotemporal chaos in terms of unstable recurrent patterns,

F Christiansen, P Cvitanović and V Putkaradze (29 apr 1996)
Nonlinearity 10, 50 (1997), chao-dyn/9606016
  Fig 1, Fig 2, Fig 3, Fig 4

Local Structures in Extended Systems

Vachtang Putkaradze (Ph.D. thesis 1997)

Universality in complex discrete dynamics

M.J. Feigenbaum (Aug 26-Sept 1, 1976)
"The Second Los Alamos workshop on Mathematics in Natural Sciences,". Los Alamos Theoretical Division Annual Report 1975-1976, pp. 98-102.
(first published report on universality in period doubling)

Exercise: Period doubling in your pocket

E. Greco (Sep 19 2006)
  matlab code - different steps of the solution,
  matlab code - webgraph only

Exercise: Period doubling in your pocket

J. Millan (Sep 19 2006)
  c/gnuplot code

Fluctuations and Irreversible Processes

L. Onsager and S. Machlup
Phys. Rev. 91 , 1505, 1512 (1953)
and the sequel [password needed]

Itô calculus

notes by A. Prügel-Bennett (June 1995)
M.J. Feigenbaum course on stochastic integration

Dynamical systems approach to 1-d spatiotemporal chaos - A cyclist's view

Yueheng Lan (Ph.D. thesis, Georgia Tech 2004)

Papers on variational periodic orbit searches

Y Lan and P Cvitanović

Implementation of the cyclist relaxation methods for the Henon and Ikeda maps

Cristel Chandre (Dec 10 2002)
  matlab code

Variational search for periodic orbits

Evangelos Siminos
  variational search fortran code

Spatiotemporally periodic solutions by variational methods,

P Cvitanović

Systematic detection of unstable periodic orbits in discrete chaotic dynamical systems

F.K. Diakonos, D. Pingel and P. Schmelcher (4 July 2000)

Lyapunov exponent for products of random matrices

Jakob Langgaard Nielsen (6 Feb 1997)

Open projects

A few suggestions for possible projects. For inspiration, see also the published projects and projects homepage.