Detection of coupling delay: A problem not yet solved.

Nonparametric detection of coupling delay in unidirectionally and bidirectionally coupled nonlinear dynamical systems is examined. Both continuous and discrete-time systems are considered. Two methods of detection are assessed-the method based on conditional mutual information-the CMI method (also known as the transfer entropy method) and the method of convergent cross mapping-the CCM method. Computer simulations show that neither method is generally reliable in the detection of coupling delays. For continuous-time chaotic systems, the CMI method appears to be more sensitive and applicable in a broader range of coupling parameters than the CCM method. In the case of tested discrete-time dynamical systems, the CCM method has been found to be more sensitive, while the CMI method required much stronger coupling strength in order to bring correct results. However, when studied systems contain a strong oscillatory component in their dynamics, results of both methods become ambiguous. The presented study suggests that results of the tested algorithms should be interpreted with utmost care and the nonparametric detection of coupling delay, in general, is a problem not yet solved.