5-6 June 2012 at the 2012 CSSS...

For the past two days the focus has been on tools for analyzing non-linear dynamical (and specifically chaotic) systems.   We have been using a program called TISEAN to do most of the analysis.  There also seem to be a number of R packages for doing non-linear times series analysis: RTisean, tsDyn, tseriesChaos, etc.

Henon Map:  

Our first task was to simply use TISEAN to generate some trajectories of the Henon map, plot them using our favorite plotting tool (at the moment I am working on improving my Python coding so I am using matplotlib) and then analyze the power spectrum (sometimes called spectral density).  TISEAN's version of the Henon map is:

x_t+1 = 1 - Ax_t² + By_t

y_t+1 = x_t

For A=0.8 and B=0, the attractor is a simple 2-cycle which means that a plot of the trajectory of the map in state space will yield two points:

 However, for A=1.4 and B=0.3, the Henon map displays chaotic dynamics:

Power Spectrum: A good place to start the analysis of times series data is to examine the power spectrum (or spectral density) of the data.   For the Henon map with A=0.8 and B=0, the attractor is a 2-cycle which implies that the dominant frequency should be 1/2.  

Note the above plot has a single "spike" at a frequency of 0.5.  What other frequencies are present in the times series generated by the Henon map?  Given that the map is a 2-cycle, in theory, there should be only a single frequency present in the data.  Although my computer can represent much smaller numbers, the smallest number that my computer can distinguish as being distinct (i.e., my machine ε) is 2.2204460492503131e-16.  The other "spikes" in the above plot are thus non-sensical results of my computer doing calculations with numbers that are too small for it to handle properly.  This is a good example of arithmetic underflow.

Given that the Henon map with A=1.4 and B=0.3 exhibits chaotic dynamics we expect that the power spectrum should exhibit spikes at all frequencies.

The Lorenz System: 

Next we want to plot some trajectories the Lorenz System and then analyze the resulting power spectrums.  Classic model of chaos developed by Edward Lorenz to model weather/climate systems.  For parameter values R=15, S=16, B=4 the system exhibits a unique fixed point attractor...

...or in 3D if you prefer:

The power spectrum for the unique fixed point attractor looks as follows:

One gets much more interesting dynamics out of the Lorenz system simply by changing R.  For parameter values R=45, S=16, B=4 the system exhibits chaos:

For R=45, the power spectrum exhibits power at all frequencies:

Python code (and the data files if you do not have TISEAN installed) for replicating the above can be found here.