Information theory in nonlinear error growth dynamics and its application to predictability: Taking the Lorenz system as an example

In nonlinear error growth dynamics, the initial error cannot be accurately determined, and the forecast error, which is also uncertain, can be considered to be a random variable. Entropy in information theory is a natural measure of the uncertainty of a random variable associated with a probability distribution. This paper effectively combines statistical information theory and nonlinear error growth dynamics, and introduces some fundamental concepts of entropy in information theory for nonlinear error growth dynamics. Entropy based on nonlinear error can be divided into time entropy and space entropy, which are used to estimate the predictabilities of the whole dynamical system and each of its variables. This is not only applicable for investigating the dependence between any two variables of a multivariable system, but also for measuring the influence of each variable on the predictability of the whole system. Taking the Lorenz system as an example, the entropy of nonlinear error is applied to estimate predictability. The time and space entropies are used to investigate the spatial distribution of predictability of the whole Lorenz system. The results show that when moving around two chaotic attractors or near the edge of system space, a Lorenz system with lower sensitivity to the initial field behaves with higher predictability and a longer predictability limit. The example analysis of predictability of the Lorenz system demonstrates that the predictability estimated by the entropy of nonlinear error is feasible and effective, especially for estimation of predictability of the whole system. This provides a theoretical foundation for further work in estimating real atmospheric multivariable joint predictability.