High-dimensional generalizations of the Lorenz system and implications for predictability

Abstract

A set of $(3N)$- and $(3N+2)$-dimensional ordinary differential equation systems for any positive integer $N$ are newly derived as high-dimensional extensions of the three-dimensional Lorenz system, and their numerical solutions are analyzed using periodicity diagrams, bifurcation diagrams, solution trajectories, and initial condition experiments. Higher-dimensional Lorenz systems extended in this manner can be considered to be closer to the original governing equations describing Rayleigh-Bénard convection in the sense that they incorporate smaller-scale motions. This study focuses on how the solution characteristics react to incremental changes in the dimension of the Lorenz system. By plotting periodicity diagrams in dimension-parameter spaces, it is shown that the parameter ranges in which the systems have chaotic solutions tend to diminish with increasing dimensions. On the other hand, for particular parameter choices that result in chaotic solutions across many dimensions, having a higher dimension does not always result in a faster or slower initial error growth. Possible implications of these results are discussed in the context of predictability.