A Review of Lorenz’s Models from 1960 to 2008

Abstract

This review presents an overview of Lorenz models between 1960 and 2008, classified into six categories based on different types of Partial Differential Equations (PDEs). These models have made significant contributions to our understanding of chaos, the butterfly effect, attractor coexistence, and intransitivity (or “almost intransitivity”) across various scientific fields. [Formula: see text][Formula: see text]Type I models include the influential Lorenz 1963 model and generalized Lorenz models. The classical Lorenz model laid the groundwork for chaos theory by revealing the sensitivity to initial conditions and chaotic behavior. Generalized Lorenz models were developed to examine the dependence of chaos on the number of Fourier modes and to illustrate attractor coexistence. [Formula: see text][Formula: see text]Type II models were derived from two-layer, quasi-geostrophic systems. These models investigated nonlinear oscillations and irregular solutions. Based on these models, in 1960, Lorenz first presented nonperiodic solutions. Type III models include the Lorenz 1960 and 1969 models, derived from a vorticity-conserved PDE. These models shed light on nonlinear oscillatory solutions, linearly unstable solutions, and the predictability estimates of the atmosphere. However, recent studies have raised doubts regarding the validity of the two-week predictability limit. [Formula: see text][Formula: see text]Type IV models, based on shallow water equations, have advanced our understanding of the coexistence of slow and fast variables. Type V models, which include models not based on specific PDEs, include the Lorenz 1984 and 1996 models used for studying intransitivity and investigating data assimilation techniques. [Formula: see text][Formula: see text]Type VI models, involving difference equations, have proven effective in demonstrating chaos and intransitivity across diverse fields. Interestingly, Lorenz’s early work in 1964 and 1969 employed the Logistic map, appearing earlier than significant studies in the 1970s. [Formula: see text][Formula: see text]In summary, the study of Lorenz models has deepened our understanding of chaos, attractor coexistence, and intransitivity (or “almost intransitivity”). Future research directions may involve exploring higher-dimensional models, utilizing advanced mathematical and computational techniques, and fostering interdisciplinary collaborations to further advance our comprehension and the prediction of capabilities regarding coexisting chaotic and nonchaotic phenomena, as well as regime changes.