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3.1 Example analysis of a time-series from a chaotic Colpitts oscillator

In this section we briefly demonstrate basic steps for anaysing a chaotic time series. The methods used will be explained ind maore detail in the following sections.

 
>> s = signal('colpitts.dat','ascii')
  s = signal object 
 
  Dlens : 6001
  X-Axis 1 :  | 
 
  Name : colpitts
  Type : 
 
  Attributes of data values : 
   | 
 
  Comment : 
 
  History : 
17-Aug-1999 15:08:24 : Imported from ASCII file 'colpitts.dat'

By entering the above command line, the overloaded constructor for class signal was called. Giving a filename as argument tells the constructor to load the datafile and convert it into a signal object. The datafile 'colpitts.dat' contains a time-series generated by an electronical Colpitts circuit that oscillates chaotically.

To plot signal s, just issue the following command :

 
view(s);

Lets find a good choice for a delay-time by using the first minimum of the auto mutual information function
 
a = amutual(s,32);
view(a);

the first minimum of the auto mutual information can be found at four. Now we need to know the minimal embedding dimension for the colpitts signal. We use Cao's method with a delay time of four, a maximal dimension of eight, three nearest neighbors and 1000 reference points.
 
c = cao(s,8,4,3,1000);
view(c);

There's a kink in the graph produced by Cao's method at three. So now do a time-delay reconstruction of the Colpitts signal with embedding dimension 3 and delay 4.
 
e = embed(s, 3, 4);
view(e);

What's the correlation dimension of the reconstructed data set ? First let's take a look at the scaling of the correlation sum versus the radius (as log-log plot).
 
view(corrsum(e, -1, 0.05, 40, 32));

Next, we use the Takens estimator for the correlation dimension. It needs basically the same input arguments as the function corrdim2.
 
>> takens_estimator(e, -1, 0.05, 40)    

ans =

    1.9483

And what about it's largest Lyapunov exponent ? To estimate the largest Lyapunov exponent, we take a look at the scaling of the prediction error.

 
view(largelyap(e, 1000, 300, 40, 2));


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