Analysis of multiple non-linear time series I need to compare multiple curves,  the dependent variable is the proportion of animals doing a particular behaviour, the independent variable is time. When I plot the values, I see that some groups show exponential behaviours, while others show sinusoidal and linear. It's obvious that the temporal dynamics are different. What statistical tests should I do to show that?
 A: You can go about this in various ways, depending on what your goal is for the comparison.
For instance, if your ultimate goal is to determine how the series are different, then you could use time series clustering. This technique would allow you to compare the series so that you can group them into several "clusters" containing series that are "similar". The clustering could be based on a set of "features" extracted from each time series (such that those features capture the underlying temporal dynamics) or it could be based on the "shapes" of the series.
Then you would characterize the time series in each cluster via a "prototypical" time series which you may be able to assign some meaning to. Describing the "prototypical" time series for your clusters would provide clues into the functional groups of time series you are working with.
If, however, you are interested in understanding why the series are different, you could use dynamic factor analysis. This technique would allow you to identify common underlying patterns among your time series. You can then "group" your time series according to which trends they load highly onto.
