*R*packages for doing non-linear times series analysis:

*RTisean*,

*tsDyn*,

*tseriesChaos*, etc.

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**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_{t}^{2}+ 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.

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**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.