Information flow between subspaces of complex dynamical systems

The quantification of information flow between subspaces in ensemble predictions for complex dynamical systems is an important practical topic, for example, in weather prediction and climate change projections. While information transfer between dynamical system components is an established concept for nonlinear multivariate time series, the specific nature of the nonlinear dynamics generating the observed flow of information is ignored in such statistical analysis. Here a general mathematical theory for information flow between subspaces in ensemble predictions of a dynamical system is developed, which accounts for the specific underlying dynamics. The results below also include potentially useful approximation strategies for practical implementation in dynamical systems with many degrees of freedom. Specific elementary examples are developed here with both stable and unstable dynamics to both illustrate facets of the theory and to test Monte-Carlo solution strategies. The theory presented here builds on recent work of Liang and Kleeman [26] on information transfer for two-dimensional systems.