ON THE NON-EQUIVALENCE OF OBSERVABLES IN PHASE-SPACE RECONSTRUCTIONS FROM RECORDED TIME-SERIES

In practical problems of phase-space reconstruction, it is usually the case that the reconstruction is much easier using a particular record ed scalar variable. This seems to contradict the general belief that a ll variables of a dynamical system are equivalent in phase-space recon struction problems. This paper will argue that, in many cases, the cho ice of a particular scalar time series from which to reconstruct the o riginal dynamics could be critical. It is argued that different dynami cal variables do not provide the same level of information (observabil ity) of the underlying dynamics and, as a consequence, the quality of a global reconstruction critically depends on the recorded variable. E xamples in which the choice of observables is critical are discussed a nd the level of information contained in a given variable is quantifie d in the case where the original system is known. A clear example of s uch a situation arises in the Rossler system for which the performance of a global vector field reconstruction technique is investigated usi ng time series of variables x, y or z, taken one at a time.