If it is assumed that the time series X affects the time series Y, is it possible to quantify the effect of X attractor, reconstructed in the phase space through Takens's theorem, on the reconstructed Y attractor?
There is Granger causality, for example, to model this causal behavior in time series. Is there any similar technique to model the causality between two attractors in the phase space ( X -> Y)?
I think you need to refine what you mean by "affects" and "causality". You can certainly calculate the cross-correlation function of one or more of the variables in the two attractors, but if they reside in independent basins, there is no reason to expect their clocks to be synchronized or even to run at the same rate. A first step might be to convert an orbit on each attractor to a scalar time series using Takens' theorem and then to calculate the cross correlation function between one and a time-shifted version of the other as a function of the time shift. Such a plot would likely have structure, but it may not be reproducible or easy to interpret.
Dear Prof. Julien Clinton Sprott ;
First of all thank you so much for taking time to answer the question and suggesting the cross-correlation function in that innovative way. I am going to simulate it on my project. However, let me explain you more about my project to clarify what I wrote about "affects" and "causality". I have working on electroencephalogram (EEG) signals (monitor the brain's electrical activities) for Alzheimer's disease detection. In one hand, quantifying the intensity of repetition of a EEG wave pattern in past of an EEG channel (X(t-tau)) on the present of the other EEG signals (Y(t-0)), known as EEG causality, can be meaningful in AD detection. On the other hand, I have found a powerful potential in EEG attractors in phase space for AD detection. Therefore, I am going to find a causal model in phase space to open a new window on AD detection. A figure is attached to clarify my idea. I would be grateful if you could help me by adding your comment.
