Influence of the singular manifold of nonobservable states in reconstructing chaotic attractors

It is known that the reconstructed phase portrait of a given system strongly depends on the choice of the observable. In particular, the ability to obtain a global model from a time series strongly depends on the observability provided by the measured variable. Such a dependency results from (i) the existence of a singular observability manifold, M(s)(obs), for which the coordinate transformation between R(m) and the reconstructed space is not defined and (ii) how often the trajectory visits the neighborhood U(M(s)(obs)) of M(s)(obs). In order to clarify how these aspects contribute to the observability coefficients, we introduce the probability of visits of M(s)(obs) and the relative time spent in U(M(s)(obs)) to construct a new coefficient. Combined with the symbolic observability coefficients previously introduced [Letellier and Aguirre, Phys. Rev. E 79, 066210 (2009)] (only taking into account the existence of M(s)(obs)), this new coefficient helps to determine the specific role played by the location of M(s)(obs) with respect to the attractor, in phase portrait reconstruction and in any analysis technique.