Examining the presence of persistent domain from time series data There are three variables, $X_t$, $Y_t$, and $Z_t$ that are dependent of each other, and I have the time series data of those variables from replicated experiments. The stochastic dynamics look quite persistent and repeat some patterns. Based on my subjective view, it looks like there is some weak attracting force that make the system persist but sometimes a balance is broken and the system will collapse (e.g., one of the variable becomes 0). The duration that a system persists can vary, but if good initial values are selected, they persist quite long. Is there any way to test (i.e., using the time series data) whether there is a weak attractor that allows the system to persist (although I cannot formally define what a weak attractor is)? 
 A: I cannot comment, so I will reply.
First: I think that first of all you should formally define what you are trying to extract from your data. What do you want to get? What's the hypothesis your are trying to validate/exclude? 
I'll make an example: let's say the data you've got represents the positions of Sun, Earth and Moon. Indeed there's an attraction between the bodies, but how would you extract it from the data? You are really compelled to make hypothesis. You could say that the force depends on distance (how?), that Newton's 3rd principle is valid, etc. But without some starting hypothesis one cannot build up a model from raw data alone. Actually, one could: somebody proved that with the same data you can build two different models that exactly reproduce the data. And of course if your model has more parameters than the data acquired, the model is just a tautology (see overfitting on wikipedia).
Second: could you post some plots of the data?
Third: what kind of experiment are you studying? What are the plausible governing equations of the system? From there you could start something...
