I am trying to find a way to classify and segment a large set of time series that each individually describe a dynamical system. I wanted to know if the following idea for doing so is a feasible method. If I used a dynamic time warping based clustering method to segment the set of time series into separate clusters, would those clusters tend to be the same in a topological sense? (i.e. they each describe dynamical systems that have the same number of attractors with the same stability types?)
I don't need perfect classification accuracy, but my assumption for trying it is that since the dynamic time warping is able to match out of phase time series, that this would still be able to detect the type of dynamical system occurring even if they are out of phase and have different initial conditions. Apologies in advance if this is poorly worded. Let me know if there are any unclear points.
Apologies, I should have been more specific. My initial idea was to apply this to primarily periodic, non linear dynamical systems with two degrees of freedom, with each time series lasting for well over the time it would take for a point to traverse around the trajectory it is on. I expect that there will be both damping and some driving force. There is stochastic noise as well. I am not sure at this point whether it would be iid white noise or not. In this case, I don't necessarily care about the bifucations/paramater based changes in the attractors themselves, but rather just that, given a dynamical system with a certain number and type of attractors, can I cluster it into groups that also have the same number and type of attractors? I don't wish to make this too open ended, but are there other techniques that would be superior to this approach? I know a little about delayed embedding algorithms for system reconstruction, and presumably I could use this to sort them into groups, but beyond that, I am pretty limited still regarding approaches.