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The predictability of the flow of bivariate cardiorespiratory data is investigated by means of the delta-epsilon method. This method was developed originally for the analysis of univariate time series in order to detect deterministic aspects of dynamics. The approach is extended here in order to detect coupling between respiratory movements and heart-rate fluctuations. Possibilities and limitations of linear and nonlinear analysis methods were illustrated by their application to simulated linear and nonlinear coupled oscillators. In this way, the importance of nonlinear analysis was indicated with regard to nonlinear dynamics of coupled systems. The physiological application of this approach focused on cardiorespiratory coordinated dynamics. The joint cardiorespiratory behavior includes significant determinism both in quiet sleep and in active sleep. The couplings between respiratory movements and heart-rate fluctuations were found to be stronger during quiet sleep than during active sleep. However, in both sleep states, nonlinear parts of couplings were found that go beyond the linear couplings quantified by the coherence function. By means of these results, it was demonstrated that the joint phase-space reconstruction of coupled quantities opens new possibilities for the evaluation of nonlinear coordinations, such as was shown by the simulated and cardiorespiratory data.