I’m originally a computer scientist; so I hope my question is not trivial. I’m working with time series and want to reconstruct the phase space from the time series based on time-lagged versions of time series. For this purpose I need to calculate $m$ and $\tau$ which are the embedding dimension and time-lag respectively.
But: Today I was seeing some experiments based on some random $m$ and $τ$ and I saw some topology in their attractor after which this question came to my mind:
Regardless of how one chooses $m$ and $\tau$, if the attractor has a meaningful topology (i.e. following a geometrically structured shape), does it mean that this attractor could capture some meaningful dynamics?
In other words assume that I calculated $m=M$ and $\tau=T$, but I get a meaningful attractor shape for another $m$ and $\tau$. How can I interpret it?