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.

Systematic comparison between the generalized Lorenz equations and DNS in the two-dimensional Rayleigh-Bénard convection

“…If we consider M > 2 and L = 1, we recover the high-order Lorenz equations 12,13 . And we can also reproduce the results of the DNS mathematically in the limits L → ∞ and M → ∞ (in practice, when L and M are sufficiently large).…”

“…solutions with at least two positive Lyapunov exponents, which was not seen in the original Lorenz equations). For a systematic comparison between the classic Lorenz equations and the higher-order extensions, Moon et al 12 thoroughly investigated the dynamical behaviors and bifurcation structures of the extended systems obtained by considering higher-order harmonics at dimensions 5, 6, 8, 9, and 11 in wide ranges of parameters, which was later generalized 13 into explicit ODE expressions for (3N)and (3N + 2)-dimensional Lorenz systems for any positive integer N.…”

“…The second issue is pertinent to the way in which the dimension is raised in the previously investigated generalizations of the Lorenz equations 12,13 , wherein the additionally incorporated higher-order harmonics are exclusively in the vertical direction of the thermal convection problem. These studies have not simultaneously considered horizontal higher-order harmonics and consequently the convection cells corresponding to very high harmonics in their generalizations may appear to have been vertically squeezed, which can lead to certain unnatural behaviors with regard to fluid convection.…”

Adaptive fixed-time synchronization of Lorenz systems with application in chaotic finance systems

“…For obtaining a fast convergence rate, the finite-time and fixed-time control techniques have been further implemented for chaotic Lorenz systems in recent years [23]- [25]. Besides, the complexity [26], high-dimension [27], and unstable periodic orbits analysis [28] of the generalized Lorenz systems have been discussed a lot. However, the signum function * is usually contained in the conventional finite-time [10]- [12], [23] and fixed-time [15,17] control techniques, which may incur the troublesome chattering phenomenon and even damage the apparatus in industrial production due to its discontinuity.…”