Physics 5413 - Chaos, fractals, and nonlinear dynamics

Thomas Vojta home
Pegasus Cluster

Physics Department


Course description



Due date Project Description Files
Aug 31, 2018 Project 1 Project files
Sep 14, 2018 Project 2 .
Sep 28, 2018 Project 3 .
Oct 12, 2018 Project 4a or Project 4b Project 4b files
Oct 26, 2018 Project 5 .
Nov 9, 2018 Project 6 .
Nov 30, 2018 Project 7 .
Dec 14, 2018 Final Project .

Examples and extra material

Note: Most of the Java applets are unsigned. If you wish to run them on your own computer, you may have to add Java security exceptions for the corresponding sites (and accept the resulting lower security).

Chapter 1 - What is chaos?

Logistic map: Time evolution
Logistic map: Bifurcation diagram
Comparison of the bifurcation diagrams of the logistic and sine maps
Renormalization of the logistic map
Lorenz attractor (see also here or here)
Double pendulum
Pendulum Lab by Franz-Josef Elmer, University of Basel, Switzerland
Bifurcation diagram of driven damped pendulum

Chapter 3 - Dynamics in state space

Chemical oscillations, part of the IDEA project at Washington State Univ.
Van der Pol oscillator, part of the Vibe oscillator simulator, see also here

Chapter 4 - Three-dimensional state space and chaos

Lorenz applet
Saddle cycle
Homoclinic and heteroclinic tangles
Horseshoe map

Chapter 5 - Iterated maps

Lyapunov exponent of the logistic map
Gaussian map

Chapter 6 - Quasi-periodicity and chaos

Periodic vs quasiperiodic motion
Sine-circle map: fixed points, limit cycles,quasiperiodic motion
Arnold tongues
Devil's staircase
Farey tree
Sine-circle map: Chaos
Sine-circle map: Lyapunov exponent

Chapter 7 - Intermittency and crises

Intermittency in the logistic map, and in the Lorenz model
another logistic map applet
Mechanism of intermittency
Crisis in the logistic map

Chapter 8 - Fractal dimensions

Coastline of Norway (Map of Europe)
Covering the coastline, fractal dimension
Koch curve, Sierpinski triangle, Sierpinski carpet
Correlation dimensions of chaotic attractors
Multifractal electric wave function (courtesy Rudolf Roemer)
Percolation applet
Simulation data for the fractal dimension of the critical percolation cluster


The Web site for plotting program Gnuplot is