Abstract
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 Rossler 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.
Highlights
Uncertainty quantification is of great theoretical importance and practical significance in investigations of complex dynamical systems
Considering realistic applications of sensitivity analysis of an aircraft system with interactions between multiple components, recently, a rigorous formalism of information transfer with respect to Shannon entropy among the components in multi-dimensional deterministic complex dynamical systems has been established in [24] based on the LK2005 formalism to deal with the uncertainty change of one component given by the other components
We present an application study of the information flows with respect to relative entropy about the Rossler system [37]:
Summary
Uncertainty quantification is of great theoretical importance and practical significance in investigations of complex dynamical systems. Considering realistic applications of sensitivity analysis of an aircraft system with interactions between multiple components, recently, a rigorous formalism of information transfer with respect to Shannon entropy among the components in multi-dimensional deterministic complex dynamical systems has been established in [24] based on the LK2005 formalism to deal with the uncertainty change of one component given by the other components. Similar to the situation with respect to absolute entropy, the evolution of the predictability dD1 dt derives from two parts: one is from the dD∗1 dt the influences of X2 and X3 according to the coupling in the joint density distribution ρ The latter is the information flows from X2 and X3 to X1, denoted by T2D,3→1. We consider the information transfer with respect to relative entropy from X2 and X3 to X1 in multi-dimensional systems firstly, which is the difference between dD1 dt and dD12\3\ dt.
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