# Abrupt Change Detection Method Based on Features of Lorenz Trajectories

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{7}

^{*}

## Abstract

**:**

## 1. Introduction

^{n}can completely describe the state of atmospheric motion. From the study of these trajectories, we can discuss the underlying mechanism of abrupt climate change in terms of geometry and dynamics. This paper presents a definition and detection method for abrupt climate changes from the perspective of ordinary differential equation trajectories.

## 2. Dynamic Definition and Detection Method of Abrupt Changes

#### 2.1. Abrupt Changes Based on Bifurcation Features

^{n}, and so the Lorenz equations are taken as the research object. This is beneficial because, first, theoretical research can be carried out to obtain an abrupt change detection method based on the trajectory evolutions, and second, the Lorenz system is highly simplified, but qualitatively represents certain features of atmospheric dynamics. The Lorenz system comprises the following set of nonlinear equations:

^{3}.

#### 2.2. Bifurcation-Type Abrupt Change Detection Method

#### 2.3. Bifurcation-Type Abrupt Change Detection Test

#### 2.3.1. Single-Index Time Series Abrupt Change Detection

^{p}

^{2}

^{n}

_{1}, T

^{p}

^{2}

^{n}

_{2}, T

^{p}

^{2}

^{n}

_{3}) indicate the points at which the time series $\left\{{S}^{xoz}{}_{i}\right\}$ changes from positive to negative (the corresponding times are approximately 1.23, 3.63, and 9.39), while the green stars (denoted as T

^{n}

^{2}

^{p}

_{1}and T

^{n}

^{2}

^{p}

_{2}) indicate the points at which $\left\{{S}^{xoz}{}_{i}\right\}$ changes from negative to positive (the corresponding times are approximately 2.76 and 4.50). For convenience, T

^{p}

^{2}

^{n}

_{1}is also used to represent the time 1.23, and T

^{p}

^{2}

^{n}

_{2}is used to represent the time 3.63.

^{p}

^{2}

^{n}

_{i}(i = 1,2,3) (note that two of these points are very close to each other, labeled as two). The times corresponding to the green stars are T

^{n}

^{2}

^{p}

_{j}(j = 1,2). Figure 3c shows the trajectory of Lorenz Equation (1) in R

^{3}; the corresponding time intervals of every directed curve in the graph are exactly the same as in Figure 3b, and the corresponding times of the pink and green stars are also the same as in Figure 3b.

^{p}

^{2}

^{n}

_{i}(i = 1,2,3), the trajectory passes from the right equilibrium area to the left equilibrium area, while at times T

^{n}

^{2}

^{p}

_{j}(j = 1,2), the trajectory passes from left to right. Figure 3c shows that the position of the trajectory can be determined by the positive and negative values of the time series $\left\{{S}^{xoz}{}_{i}\right\}$, that is, for ${S}^{xoz}{}_{i}<0$, the trajectory is in the left equilibrium region, and for ${S}^{xoz}{}_{i}>0$, the trajectory is in the right equilibrium region. However, when ${S}^{xoz}{}_{i}=0$, the trajectory transitions between the different equilibrium regions, and these are the times when the bifurcation-type abrupt changes occur.

^{R}

^{2}

^{L}; these correspond to the pink stars in Figure 3c and T

^{p}

^{2}

^{n}

_{i}(i = 1,2,3). Jumps from the left equilibrium region to the right equilibrium region are defined as SC

^{L}

^{2}

^{R}; these correspond to the green stars in Figure 3c and T

^{n}

^{2}

^{p}

_{j}(j = 1,2). We can see from Figure 3c that there are three instances of SC

^{R}

^{2}

^{L}and two instances of SC

^{L}

^{2}

^{R}in the time interval [0, 10]. Thus, our detection method did not produce any false or missed detections. This new method for detecting bifurcation-type abrupt changes is based on the dynamic characteristics of the Lorenz equation.

^{p}

^{2}

^{n}

_{i}correspond to SC

^{R}

^{2}

^{L}abrupt changes, and T

^{n}

^{2}

^{p}

_{j}correspond to SC

^{L}

^{2}

^{R}abrupt changes.

#### 2.3.2. Multi-Index Time Series Abrupt Change Detection

^{p}

^{2}

^{n}

_{i}(i = 1,2,3), and the green regions correspond to the green stars in Figure 3a, also identified by T

^{n}

^{2}

^{p}

_{j}(j = 1,2). The local map of the pink region T

^{p}

^{2}

^{n}

_{3}is given in the lower right corner.

^{p}

^{2}

^{n}

_{i}(i = 1,2,3), corresponding to times 1.13, 3.42, and 9.29, and turns from negative to positive in the green regions T

^{n}

^{2}

^{p}

_{j}(j = 1,2), corresponding to times 2.65 and 4.43. It is positive in the intervals [0, 1.13], [2.65, 3.42], and [4.43, 9.29], and negative in the intervals [1.13, 2.65], [3.42, 4.43]. Time series $\left\{{S}^{xoy}{}_{i}\right\}$ is slightly more complex. The series changes from negative to positive at times 1.14, 3.50, and 9.29 and changes from positive to negative at times 1.38, 3.95, and 9.54. Thus, in the pink transparent regions T

^{p}

^{2}

^{n}

_{i}(i = 1,2,3), corresponding to time intervals [1.14, 1.38], [3.50, 3.95], and [9.29, 9.54], $\left\{{S}^{xoy}{}_{i}\right\}$ is positive. In addition, the series changes from negative to positive at times 2.66 and 4.43 and from positive to negative at times 2.92 and 4.63. Thus, in green transparent regions T

^{n}

^{2}

^{p}

_{j}(j = 1,2), corresponding to time intervals [2.66, 2.92] and [4.43, 4.63], $\left\{{S}^{xoy}{}_{i}\right\}$ is also positive. At all other times, $\left\{{S}^{xoy}{}_{i}\right\}$ is negative.

^{p}

^{2}

^{n}

_{i}(i = 1,2). For the green transparent region T

^{n}

^{2}

^{p}

_{j}(j = 1,2), there are similar results. The times at which $\left\{{S}^{yoz}{}_{i}\right\}$ and $\left\{{S}^{xoz}{}_{i}\right\}$ are equal to zero are not consistent within the same pink or green area, with $\left\{{S}^{yoz}{}_{i}\right\}$ passing through zero slightly ahead of $\left\{{S}^{xoz}{}_{i}\right\}$.

^{3}is shown in Figure 5, and the light blue and light gray trajectories are exactly the same as in Figure 3c. The positive and negative elements of the time series $\left\{{S}^{xoz}{}_{i}\right\}$ are used to determine whether the trajectory is in the right or left equilibrium region. The times corresponding to the pink stars are when $\left\{{S}^{xoz}{}_{i}\right\}$ is equal to zero in the pink areas in Figure 4, i.e., T

^{p}

^{2}

^{n}

_{i}(i = 1,2,3), and the times corresponding to the green stars are when $\left\{{S}^{xoz}{}_{i}\right\}$ is equal to zero in the green areas in Figure 4, i.e., T

^{n}

^{2}

^{p}

_{j}(j = 1,2). The pink and green crosses indicate when the time series $\left\{{S}^{yoz}{}_{i}\right\}$ is equal to zero. This is similar to the calibration method for the pink and green stars. The pink and green trajectories are determined by the time series $\left\{{S}^{xoy}{}_{i}\right\}$: the time intervals of the pink trajectory correspond to the pink area in Figure 4 when $\left\{{S}^{xoy}{}_{i}\right\}$ is greater than zero, and the time intervals of the green trajectory correspond to the green area in Figure 4 when $\left\{{S}^{xoy}{}_{i}\right\}$ is greater than zero. It is clear that the pink stars, crosses, and trajectories indicate jumps in the trajectory of the Lorenz equation from the right equilibrium region to the left equilibrium region. Similarly, the green stars, crosses, and trajectories indicate jumps in the trajectory of the Lorenz equation from the left equilibrium region to the right equilibrium region. Thus, the occurrence of bifurcation-type abrupt changes has been detected.

#### 2.3.3. Abrupt Change Detection Effect

^{3}. We now consider the plane trajectory. First, we discuss the abrupt change detection using the time series $\left\{{S}^{xoz}{}_{i}\right\}$. Figure 6 gives the trajectory in the xoz plane over the time interval [0, 2000]. The curves and points are colored as in Figure 5a. There are seven blue tracks on the upper right side of the left equilibrium region, corresponding to the seven green stars and seven pink stars, and there are eight gray tracks on the upper left side of the right equilibrium region, corresponding to eight green stars and eight pink stars. Thus, this seems to indicate 30 abrupt changes, but these are actually false abrupt changes. The reasons for such false detections may be that the trajectories in these locations are close to straight lines, and when calculating the time series $\left\{{S}^{xoz}{}_{i}\right\}$, machine storage limitations and calculation errors may cause nonzero values to be calculated as zero values, giving a false detection. The abrupt change points are basically clustered on the pink and green star lines. There are 602 SC

^{R}

^{2}

^{L}abrupt changes and 601 SC

^{L}

^{2}

^{R}abrupt changes, with a false detection rate of 2.5%. Note that our method cannot determine whether there are any missed detections.

^{R}

^{2}

^{L}and SC

^{L}

^{2}

^{R}abrupt change processes can be detected (as shown by the pink and green trajectories in the figure). In Figure 8b, the SC

^{R}

^{2}

^{L}and SC

^{L}

^{2}

^{R}abrupt change processes are not easy to distinguish, and the position of the trajectory cannot be accurately determined. Similar conclusions can be reached for Figure 8c,d. The reason for this detection error is that the trajectory rotates clockwise in both the left and right equilibrium regions. Only in the green region in Figure 8 does it rotate counter-clockwise. Therefore, it is difficult to detect the position and abrupt change type, but the spatial location of the abrupt change must be in the green area in Figure 8d.

## 3. Conclusions and Implications

^{n}is the corresponding trajectory. The abrupt change detection method proposed in this paper has been applied to trajectories in the normed linear space R

^{3}; therefore, it can theoretically be used on trajectories in any normed linear space R

^{n}. Consider the case in which T, u, v, w, p, and ρ denote time series of temperature, wind speed (three components of velocity), pressure, and density, respectively. The time series $\left\{{S}^{xoy}{}_{i}\right\}$ associated with different meteorological elements can be constructed from Equation (4), where x and y are two of T, u, v, w, p, and ρ. By identifying the points at which the time series $\left\{{S}^{xoy}{}_{i}\right\}$ is equal to zero, it is possible to determine the times at which abrupt climate changes occur. It is important to note that, although this method can theoretically be applied to climatic change, its application to real atmospheric data requires further investigation in the future. Once the dimension becomes very high, the discreteness of the observation data may cause some technical problems. In summary, the proposed abrupt change detection method has been established based on the dynamic significance of the system trajectories, and is able to detect abrupt changes in multi-dimensional time series.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Trajectory of the Lorenz equation and y component. (

**a**) Trajectory of the Lorenz equation, (

**b**) y component of one periodic trajectory, (

**c**) y component of semi-periodic trajectory, (

**d**) y component of transition trajectory from left equilibrium region to right equilibrium region, and (

**e**) y component of transition trajectory from right equilibrium region to left equilibrium region.

**Figure 3.**Area index time series and trajectory of the Lorenz equation. (

**a**) Area index time series $\left\{{S}^{xoz}{}_{i}\right\}$(

**b**) Trajectory of the Lorenz equation in coordinate plane xoz. (

**c**) Trajectory of the Lorenz equation.

**Figure 5.**Trajectory of the Lorenz equation. (

**a**) Trajectory in xoz coordinate plane. (

**b**) Trajectory in yoz coordinate plane. (

**c**) Trajectory in xoy coordinate plane. (

**d**) Trajectory of the Lorenz equation.

**Figure 8.**Trajectory of the Lorenz equation in xoy coordinate plane. (

**a**) Integral interval [0, 30]. (

**b**) Integral interval [0, 100]. (

**c**) Integral interval [0, 1000]. (

**d**) Integral interval [0, 2000].

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**MDPI and ACS Style**

Da, C.; Shen, B.; Song, J.; Xaiwu, C.; Feng, G.
Abrupt Change Detection Method Based on Features of Lorenz Trajectories. *Atmosphere* **2021**, *12*, 781.
https://doi.org/10.3390/atmos12060781

**AMA Style**

Da C, Shen B, Song J, Xaiwu C, Feng G.
Abrupt Change Detection Method Based on Features of Lorenz Trajectories. *Atmosphere*. 2021; 12(6):781.
https://doi.org/10.3390/atmos12060781

**Chicago/Turabian Style**

Da, Chaojiu, Binglu Shen, Jian Song, Cairang Xaiwu, and Guolin Feng.
2021. "Abrupt Change Detection Method Based on Features of Lorenz Trajectories" *Atmosphere* 12, no. 6: 781.
https://doi.org/10.3390/atmos12060781