# Lorenz-63 Model as a Metaphor for Transient Complexity in Climate

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## Abstract

**:**

## 1. Introduction

## 2. Duration of Transients and Its Relationship to Trajectory-Averaged Local Stability Multipliers

_{0}, y

_{0}, z

_{0}) in the system’s phase space:

^{T}J, where J is the finite-time Jacobian matrix whose evolution satisfies the tangent linear equations) to tag the trajectories produces essentially identical results (not shown). Thus, the longest transients are naturally associated with trajectories that tend to “travel sideways” along the dynamical slopes of the Lorenz-system “global” landscape, rather than going straight downhill toward the asymptotic attractor.

## 3. Types of Transient Behavior

#### 3.1. Sensitivity of Transients to Initial Conditions

#### 3.2. Geometric Complexity of Transients

## 4. Summary and Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Distribution of transient times (color shading) to the Lorenz attractor for initial conditions on the sphere S with the radius a = 150 centered at the point (0, 0, 24); the center of this sphere was chosen to be close to the asymptotic time mean of trajectories simulated by the Lorenz model (1) with $\sigma =10$, r = 28, h = 1, and b = 8/3. The trajectory initialized on S was considered transient until its first entry into the rectangular cuboid region B bounded by x-, y- and z-ranges of (–19, 19), (–25, 25) and (4, 46), respectively. The Lorenz attractor for the model parameters considered is located within this region. The four figure panels display the same quantity, but from different view angles. Comment: Note non-uniformity of the transient-time distribution, with a spiraling belt of relatively long durations.

**Figure 2.**Distribution of the averaged maximum local stability multipliers ${\mathsf{\Lambda}}_{\mathrm{max}}$ computed over the transient portion of trajectories (before first entry into the attractor region B) initialized on the same sphere S as in Figure 1. The same four orientations as in Figure 1 are shown. Comments: Most of the trajectories exhibit positive local-stability-multiplier averages. There is a clear correspondence between the spiraling belt of largest averaged local stability multipliers and that of longest transient duration times in Figure 1.

**Figure 4.**(

**a**,

**b**) An example of a “ghost” transient attractor in the simulation of the Lorenz model (1) with $\sigma =10$, r = 28, h = 1, and b = 8/3; (

**c**,

**d**) a trajectory of the Lorenz system with parameters (σ, r, h, b) stretched by ε

^{−1}= 3. See text for details. Comments: The transient trajectory in (a, b) does not simply spiral toward the attractor, but exhibits a complex path reminiscent of that on asymptotic attractor. The path of the rescaled Lorenz system in (c, d) shares geometrical similarity with the transient path in (a, b).

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Kravtsov, S.; Tsonis, A.A. Lorenz-63 Model as a Metaphor for Transient Complexity in Climate. *Entropy* **2021**, *23*, 951.
https://doi.org/10.3390/e23080951

**AMA Style**

Kravtsov S, Tsonis AA. Lorenz-63 Model as a Metaphor for Transient Complexity in Climate. *Entropy*. 2021; 23(8):951.
https://doi.org/10.3390/e23080951

**Chicago/Turabian Style**

Kravtsov, Sergey, and Anastasios A. Tsonis. 2021. "Lorenz-63 Model as a Metaphor for Transient Complexity in Climate" *Entropy* 23, no. 8: 951.
https://doi.org/10.3390/e23080951