Information Transfer with Respect to Relative Entropy in Multi-dimensional Complex Dynamical Systems

In this paper, a rigorous formalism of information transfer with respect to relative entropy or Kullback-Leiber divergence within a multi-dimensional deterministic dynamical system is established. It is derived from the mechanism that the governance of the predictability change could come from the evolution itself and a transfer of the evolutions of multiple components for a given component. The presented formalism of three-dimensional systems and its several generalizations in high-dimensional systems provide a precise quantification of transfers among variables in complex dynamical systems, with which some properties are explored and given. These results of information transfers are different from that with respect to Shannon entropy in multi-dimensional systems, due to a minus sign which reflects the opposite notion of predictability vs. uncertainty. Explicit formulas are demonstrated and verified in the $\text{R}\ddot {o}$ ssler system and a four-dimensional system. These studies can be used to investigate the propagation of uncertainties and perform the dynamic sensitivity analysis statistically. The simulation results suggest that the generalized formalisms provide more underlying information about multi-dimensional dynamical systems compared with currently existing methods. It is beneficial for prediction and control of systems better, with broad application prospects in many fields.