A rigorous formalism of information transfer between dynamical system components. II. Continuous flow

The transfer of information between dynamical system components is formalized with causality faithfully represented. In a continuous system with many components, information transfer is a mechanism controlling the marginal entropy evolution of the target component. It is measured by the rate of entropy thus transferred, which is obtained through freezing the source component instantaneously, and comparing the entropy increases between the original system and the so modified system. The resulting transfer measure is consistent with our earlier 2D formalism derived in Liang and Kleeman [X.S. Liang, R. Kleeman, Information transfer between dynamical system components, Phys. Rev. Lett. 95 (24) (2005) 244101] using different methods; it also possesses a property of unidirectionalism which has been emphasized by Schreiber [T. Schreiber, Measuring information transfer, Phys. Rev. Lett. 85 (2) (2000) 461-464]. We apply our formalism to a two-mode (four-dimensional) truncated Burgers-Hopf system. No significant information exchange is identified between the four components, save for a transfer from the cosine direction of mode 2 to the sine direction of mode 1. This transfer occurs continuously and at a nearly constant rate. The present work should serve as a starting point for the development of a rigorous dynamics-free formalism for the information transfer of multivariate time series.