# Revisiting Lorenz’s Error Growth Models: Insights and Applications

## Definition

**:**

## 1. Introduction

## 2. Lorenz’s Equations

#### 2.1. Lorenz’s Error Growth Models

#### 2.2. Nonlinear and Linear Relative Growth Rates

- (1)
- An initial linear increase driven by the equation’s linear term;
- (2)
- A reduction in growth rate moderated by the equation’s nonlinear component as the error expands; and
- (3)
- A leveling off at a constant error level.

#### 2.3. The Logistic ODE and Continuous Dependence on ICs (CDIC)

#### 2.4. The Logistic Map and Chaotic Solutions

#### 2.5. The Logistic ODE and the Non-Dissipative Lorenz Model (NLM)

#### 2.6. A Variant of the Logistic ODE for a Finite Predictability Horizon

- (1)
- When $t\to \infty $, $y\left(t\right)\to 1$. This is a stable asymptotic value.
- (2)
- When $t\to -\infty $, $y\left(t\right)\to -b/\sigma $. This represents an unstable asymptotic value, which is less than zero.
- (3)
- When $b=0$, $y\left(t\right)=\frac{1}{1-\sigma r{e}^{-\sigma t}}$. This solution is the same as Equation (2) within the original logistic ODE.

## 3. Concluding Remarks

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The solution of u to the logistic ODE in Equation (1). (

**a**) Four points, A, B, C, and D, indicate locations for the values of u = 0.001, 0.1, 0.9, and 0.999, respectively. (

**b**) The bottom panel indicates that it requires more than 4 time units for u to increase from 0.994 to 1.

**Figure 3.**Solutions of the logistic map in Equation (13). The three panels display a steady-state, periodic, and irregular solution, respectively.

**Figure 4.**Panels (

**a**–

**c**) display the sigmoid, hyperbolic tangent, and hyperbolic secant squared functions, respectively, indicating the relationship between the Lorenz error growth model (i.e., the Logistic ODE) and the non-dissipative Lorenz 1963 model. See main texts for details. The solution in panel (

**a**) is $u\left(t\right)$ = 1/(1 + $exp$($-\sigma $(t − 4))) with $u\left(4\right)=1/2$ for a comparison.

**Figure 5.**Solutions of the modified logistic ODE in Equation (25) with an additional parameter b under the initial condition $y(t=0)={y}_{0}=0.001$. (

**a**) Solutions for three different values of b. The solution with $b=0$ closely matches the solution of the original logistic ODE in Equation (2). (

**b**) Time intervals between the initial value ${y}_{0}$ and $y=0$ for three solutions curves, used for illustrating the impact by continuously reducing the initial error to $y=0$. The value ${y}_{0}$ signifies the error inherent in the current technology, while $y=0$ represents the error in an ideal scenario.

**Table 1.**Comparative analysis of the logistic ODE, logistic map, and non-dissipative Lorenz model (LM). The text discusses how the first model serves as a basis for deriving the second and third equations.

Name | Equation | Remarks |
---|---|---|

Logistic ODE | $\frac{du}{dt}=\sigma u(1-u)$ | Equation (1) |

Logistic Map | ${Y}_{n+1}=\rho {Y}_{n}(1-{Y}_{n})$ | Equation (15) |

Non-dissipative LM | $\frac{{d}^{2}Z}{d{t}^{2}}-{\sigma}^{2}Z+6\sigma {Z}^{2}=0$ | Equation (21) |

**Table 2.**A nonlinear second-order ODE representing four different physical solutions. This ODE appears as the Korteweg-de Vries (KdV) equation in the traveling wave coordinate, the non-dissipative Lorenz 1963 [19] (L63) model, the KdV-SIR equation, and the inviscid Pedlosky model. It is used to study solitary waves, homoclinic orbits, epidemic waves, and nonlinear baroclinic waves, respectively. For the inviscid Pedlosky model, which is mathematically identical to the X component of the non-dissipative L63 model (e.g., Shen 2021 [35]), the second order ODE of a new variable Q ($Q={P}^{2}$) is the same as the Z component of the non-dissipative L63 model. ${D}_{0}$ is a constant. (Table reproduced from Paxson and Shen (2022) [13]. With permission from copyright owner International Journal of Bifurcation and Chaos).

System’s Name | Equations | References |
---|---|---|

KdV Equation | ${f}^{\prime \prime}+3{f}^{2}-cf=0$ | Equation (14) in [13] |

Non-dissipative Lorenz Model | ${Z}^{\prime \prime}+3\rho {Z}^{2}-4\rho rZ=0$ | Equation (12a) |

${X}^{\prime \prime}+\frac{{X}^{3}}{2}-\sigma \rho X=0$ | Equation (12b) in [13] | |

KdV-SIR Equation | ${I}^{\prime \prime}+\left(\frac{3{\sigma}^{2}}{2{I}_{max}}\right){I}^{2}-{\sigma}^{2}I=0$ | Equation (17a) in [13] |

Inviscid Pedlosky Model | ${P}^{\prime \prime}-({D}_{0}+1)P+{P}^{3}=0$ | Pedlosky (1971) [34] |

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**MDPI and ACS Style**

Shen, B.-W.
Revisiting Lorenz’s Error Growth Models: Insights and Applications. *Encyclopedia* **2024**, *4*, 1134-1146.
https://doi.org/10.3390/encyclopedia4030073

**AMA Style**

Shen B-W.
Revisiting Lorenz’s Error Growth Models: Insights and Applications. *Encyclopedia*. 2024; 4(3):1134-1146.
https://doi.org/10.3390/encyclopedia4030073

**Chicago/Turabian Style**

Shen, Bo-Wen.
2024. "Revisiting Lorenz’s Error Growth Models: Insights and Applications" *Encyclopedia* 4, no. 3: 1134-1146.
https://doi.org/10.3390/encyclopedia4030073