This entry examines Lorenz’s error growth models with quadratic and cubic hypotheses, highlighting their mathematical connections to the non-dissipative Lorenz 1963 model. The quadratic error growth model is the logistic ordinary differential equation (ODE) with a quadratic nonlinear term, while the cubic model is derived by replacing the quadratic term with a cubic one. A variable transformation shows that the cubic model can be converted to the same form as the logistic ODE. The relationship between the continuous logistic ODE and its discrete version, the logistic map, illustrates chaotic behaviors, demonstrating computational chaos with large time steps. A variant of the logistic ODE is proposed to show how finite predictability horizons can be determined, emphasizing the continuous dependence on initial conditions (CDIC) related to stable and unstable asymptotic values. This review also presents the mathematical relationship between the logistic ODE and the non-dissipative Lorenz 1963 model.
The study offers a concise overview of the mathematical framework for Lorenz error growth models (Lorenz 1969a [
1]) and their connections to other relevant systems (e.g., the non-dissipative Lorenz model, Shen 2018, 2020 [
2,
3]). In the 1960s, Lorenz utilized the first-order ordinary differential equations (ODEs) incorporating quadratic or cubic nonlinear terms to study root mean square (RMS) errors, aiding in the estimation of doubling times and thus predictability horizons (Lorenz 1969a, b [
1,
4]; Shen et al., 2024 [
5]). The model with the quadratic term is commonly referred to as the logistic ODE. In contrast, the error growth model with a cubic term can be transformed into the logistic ODE (with a quadratic term). It is important to distinguish this error growth model from Lorenz’s widely recognized 1969 multiscale model in meteorology (Lorenz 1969c [
6]; Shen et al., 2024 [
5]).
Since the 1960s, the logistic ODE, the continuous form of the logistic equation, has been extensively used to assess RMS forecast errors in ensemble forecasts (e.g., Lorenz 1969a, 1982, 1996 [
1,
7,
8]; Nicolis 1992 [
9]; Kalnay 2003 [
10]; Zhang et al. 2019 [
11]). Historically, the logistic ODE has also been used to analyze population growth, assuming that the growth rate of a population is proportional to both the existing population and the available resources (represented by the growth rate
𝜎in Equation (
1)). The logistic ODE-based model offers a more realistic portrayal of population growth compared to simple exponential models. This model has also been adapted to study the dynamics of recovered individuals during the COVID-19 pandemic (Postnikov, 2020 [
12]; Paxson and Shen 2022 [
13]).
The logistic map, a discrete form of the logistic equation, stands as the most straightforward model for demonstrating chaos, as recognized in the works of Lorenz (1964, 1969d) [
14,
15], May (1976) [
16], and Li and Yorke (1975) [
17]. This map exhibits both periodic and chaotic behaviors, heavily influenced by a system parameter that acts as a forcing term. Initially introduced by Lorenz in 1964, the map was used to describe the transition from regular to irregular solutions in dynamic systems. Robert May (1976) [
16] utilized this map to investigate population dynamics in biological contexts, while Li and Yorke (1975) [
17] employed it to demonstrate the concept of “Period Three Implies Chaos”, where the term “chaos” was initially introduced to describe irregular solutions. Consequently, the logistic map has become an influential tool in studying chaos and complexity within dynamical systems and was recognized by Stewart (2013) [
18] as one of the 17 equations that changed the world.
This article explores both the continuous and discrete versions of the logistic equation, providing a mathematical perspective on Lorenz models. It highlights a crucial, yet often overlooked, aspect of continuous dependence on initial conditions (CDIC). This concept is demonstrated by showing how the predictability horizon, as derived from the logistic ODE, can be extended progressively by refining the initial conditions. The structure of the study is as follows: After presenting the equations for the Logistic ODE, this entry analyzes its error growth rates and its linkage to the Logistic map. The Logistic ODE and map are employed to illustrate CDIC and irregular solutions, respectively. This entry also explores the connection between the logistic ODE and the non-dissipative Lorenz model (Lorenz 1963 [
19]; Shen 2020, 2023 [
3,
20]), and finally discusses the finite predictability horizons in a variant of the logistic ODE. Concluding remarks are presented at the end.
This entry is adapted from the peer-reviewed paper 10.3390/encyclopedia4030073
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