Embedding theorems for non-uniformly sampled dynamical systems

The embedding theorem of Takens and its extensions have provided the theoretical underpinning for a wide range of investigations of time series derived from nonlinear dynamical systems. The theorem applies when the dynamical system is sampled uniformly in time. There has, however, been increasing interest in situations where observations on the system are not uniform in time, and in particular where they consist of a series of inter-event ('interspike') intervals. Sauer has provided an embedding theorem for the case where these intervals are generated by an integrate-and-fire mechanism. Here we prove several embedding theorems pertaining to non-uniform sampling. We consider two situations: in the first, the observations consist of the values of some function on the state space of the system, with the times between successive observations being given by another function-the sampling interval function; in the second, the sampling times are generated in the same way, but now the observations consist only of the intersample intervals. We prove embedding theorems both when the sampling interval function is allowed to be fairly general, and when it corresponds to integrate-and-fire sampling. We point out that non-uniform sampling might lead to better reconstructions than uniform sampling, for certain kinds of time series.