Trajectories construction ...
Currently I am working with: Lorenz, Rossler and Hennon Systems
And here it is a portrait of the atractor:
And here it is a portrait of the atractor:
This is a Poincare Section at x=10 plane (only the crossings from right to left are taken). The section was taken while 10 different trajectories was moving along the attractor.
Points are ordered by their piercing order. In this way If the distance between the point i and the i+1 is smaller than a given threshold we can say that this close return is due to the existence of a close UPO with peridod 1. In the image shown below the treshold was set to 1. I have found that choosing this threshold is far form easy since as larger the period I am looking for the smaller the threshold should be. The problems about this issue are further discussed ahead.
This is perhaps the most difficult part of this problem since it involves to issues hard to automate. The first one is the selection of the threshold which will define when a pierce is "close" to another. And the second one is the clustering of the resulting groups. In the image above seems to be clear that there are 6 groups. Easy task for the human eye-brain combination, however it isn't so easy for a dumb computer. The clustering of groups can be done using a popular algorithm called k-means which will be explaned later.
Why is this so hard? ... Choosing the threshold is complicated since it can't be too small because some of the UPOs may not be reveled and it can't be to large because all the points will look as close returns. Here is an example: If we use the example shown before but we set the threshold to 0.1 instead of 1 we get only 3 groups, and if we make it 2.5 then a lot of points will get into the picture making the clustering difficult and probably inaccurate. To solve this problem I've designed a program called UPOExplorer which will allow me to explore the behavior of the threshold and some other things related to clustering before moving on to the automation.
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Clusters Finding: k-Means algorithm ...
You can find a great explanation of k-mens algorithm at
http://www.engr.sjsu.edu/~knapp/HCIRDFSC/C/k_means.htm
I will show some results of the algorithm performance...
Classifying three groups (the easy case ...)
In this case the algorithm permorm fine most of the times
However some time it does crazy stuff like this ...
As can be seen on the image the algorithm found 3 groups which satisfys the convergence conditions, however is obviously wrong.
images and resutls