Background

We have five exercises on discrete maps. Before working on this one, you should first do the Logistic Map exercises.

In this exercise, we use scaling and renormalization--group methods to study the onset of chaos. There are several routes by which a dynamical system can start exhibiting chaotic motion: we study the period doubling cascade, first extensively investigated by Feigenbaum.

Learning Goals

Science: You will learn about Chaos, discrete maps, and fractal dimensions.

Computation: You will learn about iterative algorithms, and techniques to explore asymptotic behaviors.

Procedure

• Download and have a quick glance at Brief notes on universality, fractal dimensions, and the renormalization group.
• Download the file ChaosLyapunovHints.py and rename it "ChaosLyapunov.py".
• Open this file in a text editor (kate or emacs -- we recommend against using kedit) or load it into the IPython dashboard (started with ipython notebook --pylab inline.
• Open a terminal window, move to the correct directory and start ipython, or click on the notebook in the dashboard. You will find it convenient to start python with the --pylab flag -- ie type ipython --pylab or ipython notebook --pylab.
• Follow the directions in Period Doubling Exercise.