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.

Monday 4 June 2012 at the 2012 CSSS...

Today's lecture, given by Prof. Liz Bradley, focused on the basics of non-linear (mostly chaotic) dynamics in both discrete and continuous time.  Much (all?) of the material in the lecture I had already encountered before in my own reading, but it was nice to get a refresher.  What follows is my summary of our discussion of the dynamical properties of that classic example of chaotic dynamics in discrete time, the logistic map.  The seminal reference for the logistic map is probably the 1976 Nature paper by Robert May.

The Logistic Map is an innocent looking non-linear equation:

X_t+1 = r X_t (1 - X_t)

where the state space of the model is the unit interval [0, 1] and r is a parameter that varies on (0, 4].  

Some trajectories of the logistic map for various values of r:

If  0 < r ≤ 1, then the dynamics are trivial: the model will converge to X = 0 no matter the initial condition.  Suppose the r = 2.  In this case, the dynamics of the model are also pretty boring: convergence to a unique fixed point.  No matter the initial condition, if r = 2, all trajectories of the logistic map will converge (quite quickly) on a unique steady state value of 1/2.

Now suppose that r = 2.919149.  In this case, the result is still convergence to a unique fixed point, however the dynamics are more interesting: the trajectories now exhibit dampened oscillations.

For values of r satisfying 3 < r < 3.45 (roughly!), you get a 2-cycle:

For r = 3.8285, one gets the famous 3-cycle which is one of the generally accepted indicators of chaotic dynamics (note I have changed the initial condition from 0.2 to 0.5 to eliminate the transient and thus making the cobweb diagram a bit cleaner):

Finally, for r = 4 we get an example of chaotic dynamics.  Note that the deterministic trajectory for the logistic map looks incredibly "random."

3D plot of the phase space for the logistic map:
One of the reasons why models with chaotic dynamics, such as the logistic map, exhibit sensitive dependence to initial conditions, is that such maps repeatedly "stretch and fold" the state space over which they are defined.  While 2D phase plots give a sense of how the logistic map "stretches" the state space,  a 3D phase plot is a really cool way to see how the logistic map "folds" the state space.

If you are interested, the Python code to replicate the above graphics can be found here.  I am still working on the code for the bifurcation diagram, estimating Feigenbaum's constant, and for calculating Lyapunov exponents.  

Santa Fe bound...

Stopping over in Alexandria, VA to visit friends on the way to Santa Fe for the 2012 CSSS.  Posting will be more frequent (I hope!) over the next month.