  Symbolic relative entropy, an efficient nonlinear complexity parameter
measuring probabilistic divergences of symbolic sequences, is proposed in our
nonlinear dynamics analysis of heart rates considering equal states. Equalities
are not rare in discrete heartbeats because of the limits of resolution of
signals collection, and more importantly equal states contain underlying
important cardiac regulation information which is neglected by some chaotic
deterministic parameters and temporal asymmetric measurements. The relative
entropy of symbolization associated with equal states has satisfied nonlinear
dynamics complexity detections in heartbeats and shows advantages to some
nonlinear dynamics parameters without considering equalities. Researches on
cardiac activities suggest the highest probabilistic divergence of the healthy
young heart rates and highlight the facts that heart diseases and aging reduce
the nonlinear dynamical complexity of heart rates.
