Evolutionary Analysis of Air Traffic Situation in Multi-Airport Terminal Areas

Abstract

As the demand for air transportation surges, issues like flight conflicts and air-route congestion within multi-airport terminal areas have grown progressively more serious. Analyzing the evolution of air traffic situations in these areas can effectively enhance the air traffic’s early-warning capability, reduce flight conflicts, and alleviate air-route congestion. This paper proposes a method for analyzing the evolution of air traffic situations in multi-airport terminal areas based on flight segment–flight state interdependent network. First, a flight segment–flight state interdependent network model is established. This interdependent network model consists of an upper-layer flight state network, a lower-layer air-route network, and coupling edges. The upper-layer network is constructed with aircraft as nodes and flight conflicts between aircraft as edges. The lower-layer network takes air-routes as nodes and the connection relationships between air-routes as edges. The inter-layer coupling edges are determined by judging the relationship between aircraft and air-routes. If an aircraft is on a certain air-route, there exists a coupling edge between the aircraft node and the air-route node. On this basis, by comprehensively considering three network indicators, namely node degree, weighted clustering coefficient, and node strength, the overall air traffic situation value is obtained. Finally, experimental verification and analysis were conducted in an actual flight scenario of a multi-airport terminal area in the Guangdong–Hong Kong–Macao Greater Bay Area. The results show that the proposed method can accurately reflect the air traffic situation. The time-series analysis of the situation evolution reveals that the evolution process has chaotic characteristics.

1. Introduction

The multi-airport terminal area is an integrated air traffic control area composed of multiple airports and their surrounding airspace, which can make up for the deficiencies of single-airport terminal areas in terms of capacity, coverage, etc. In recent years, with the rapid development of the air transportation industry, the problems of flight conflicts and air-route congestion in the multi-airport terminal area have become increasingly prominent. In-depth research on the evolution pattern of its air traffic situation is conducive to reducing the risk of conflicts, alleviating air-route congestion, and improving the utilization efficiency of air resources. The evolution of air traffic refers to the continuous process of change and development of the air traffic system over time. The laws of evolution are the universal, recurring, and predictable internal rules manifested in this process.

Currently, research on air traffic situation analysis mainly focuses on the following two aspects: one is situation analysis based on controller cognition, and the other is situation analysis based on aircraft flight states. In addition, there are also meteorological factors; system infrastructure and capacity; integration with automatic prediction tools (machine learning and simulation models); and regulatory and operational aspects (minimum separation and traffic rules). In the field of controller cognition, Histon et al. [1] were the first to propose the concept of cognitive complexity, and Fenza et al. [2] subsequently used cognitive methods to construct a situation awareness model. However, most of this kind of research centers around controller workload assessment and is significantly influenced by human factors.

When conducting a situation assessment based on the flight states of aircraft operating on air-routes, the main factors considered are the number of aircraft and their positions and headings. In recent years, many scholars have carried out situation assessment work using complex network methods based on these factors: Wang Hongyong et al. [3,4,5] constructed a two-dimensional weighted conflict network with aircraft as nodes and flight connections as edges, and measured the air traffic situation through indicators such as the node degree and connectivity rate; Wu Minggong et al. [6,7,8] established a flight-state network, described the situation using network topology indicators, and monitored the air traffic dynamics in combination with an independent principal component analysis to reduce the interference of subjective factors.

The above-mentioned complex network research is all limited to topological structure analysis under the single-layer network. However, the air traffic system is essentially a system with multi-factor intersections. Therefore, a multi-layer network can better describe the complexity of air traffic, and the proposal of the interdependent network theory provides a brand-new idea for solving this problem. S. V. Buldyrev et al. [9] first proposed the interdependent network theory; J. Shao et al. [10] constructed a one-to-many interdependent network model based on the coupling mode of each layer of the network; and Wang Xinglong et al. [11,12,13] established a static air traffic interdependent network model with airports, air-routes, and control sectors as sub-networks. This paper constructs an interdependent network by combining the air-route structure of a multi-airport terminal area and information about aircraft in the air, and analyzes the air traffic situation through the interdependent network model.

In the field of situation evolution research, Jiang and Yang used simulation software to build an operation scenario and explored the evolution law of congestion conditions in the terminal airspace from both temporal and spatial perspectives [14,15]. Time-series analysis is an important means to reveal the evolution mechanism of complex systems and predict the future states of systems [16].

From the perspective of air-route operation, this paper proposes a research method for analyzing the evolution of air traffic situations in multi-airport terminal areas based on the air-route–flight state interdependent network. It analyzes three key network node indicators that reflect the situation, namely the degree of air-route nodes, node strength, and weighted clustering coefficient, and then analyzes the evolution process of the overall air traffic situation.

2. Construction of the Interdependent Network

The flight segment–flight state interdependent network model is composed of an upper-layer aircraft network and a lower-layer air-route network connected by coupling edges, as shown in Figure 1. The “routes” mentioned throughout the article refer to the “standard inbound routes” and “standard outbound routes”.

The flight-state network serves as the upper-layer network of the interdependent network model. We constructed the flight-state network with the aircraft operating on the air-routes as the nodes of this network and the conflict relationships between the aircraft as the edges. Article 251 of the “Rules for Civil Aviation Air Traffic Management” of the Civil Aviation Administration of China (CAAC) stipulates that under procedural control conditions, the minimum horizontal separation between two aircraft is 20 km. If the horizontal separation is less than the specified value and there is no vertical separation (≥300 m). Therefore, we believe that if the horizontal distance between two aircraft is less than 20 km and the vertical separation is less than 300 m, there is a potential flight conflict between the two aircraft. When the distance between two aircraft is closer, the degree of conflict between them is higher, so the corresponding weight also increases. To avoid an excessively large weight, we restricted it to the range between 0 and 1. That is, when the distance between two aircraft approaches zero, the weight tends to 1; when the distance between two aircraft approaches 20 km from the left, the weight tends to 0; and when the distance between two aircraft is greater than or equal to 20 km, the weight is 0. The calculation formula for the edge weight is as follows:

$${\mathrm{w}}_{\mathrm{i}\mathrm{j}}=\left\{\begin{array}{lll}1-{\mathrm{e}}^{1-{\displaystyle \frac{20}{{\mathrm{D}}_{\mathrm{i}\mathrm{j}}}}}& & 0<{\mathrm{D}}_{\mathrm{i}\mathrm{j}}<20\\ 0& {\mathrm{D}}_{\mathrm{i}\mathrm{j}}\ge 20& \end{array}\right.$$

(1)

Among them, i represents aircraft node i, j represents aircraft node j, and D_ij is the distance between aircraft node i and aircraft node j.

The air-route network serves as the lower-layer network of the interdependent network model. We constructed the air-route network with air-routes as the nodes and the connection relationships between air-routes (the existence of waypoints between two air-routes) as the edges. Therefore, the air-route network is an undirected and unweighted network. “The existence of waypoints between two routes” refers to the common waypoints between two different routes.

If an aircraft is flying on a certain air-route, there is an edge between the aircraft node and the air-route node, and this edge is called an inter-layer coupling edge. The inter-layer coupling edges connect the air-route nodes in the air-route network and the aircraft nodes in the flight-state network. A schematic diagram of the inter-layer coupling edges is shown in Figure 2.

In Figure 2, the red line segments represent the coupling edges. Aircraft No. 1–2 are flying on Air-route No. 4, Aircraft No. 3 is on Air-route No. 3, Aircraft No. 4–5 are on Air-route No. 2, and Aircraft No. 6 is on Air-route No. 1. Thus, it can be known that there are edges connecting the nodes of Aircraft No. 1–2 to the node of Air-route No. 4, edges connecting the node of Aircraft No. 3 to the node of Air-route No. 3, edges connecting the nodes of Aircraft No. 4–5 to the node of Air-route No. 2, and edges connecting the node of Aircraft No. 6 to the node of Air-route No. 1.

The weight setting of the coupling edges takes into account the proportion of the flight distance of an aircraft on an air-route to the total length of that air-route and the flight conflicts between aircraft. The conflict multiplier is set as K (the setting of K depends on the flight scenario). The calculation formula for the weight of the coupling edges is as follows:

$${\mathrm{w}}_{\mathrm{i}\mathrm{m}}=\left\{\begin{array}{ll}({\mathrm{D}}_{\mathrm{m}}-{\mathrm{d}}_{\mathrm{i}\mathrm{m}})/{\mathrm{D}}_{\mathrm{m}} \mathrm{N}\mathrm{o} \mathrm{f}\mathrm{l}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{t}& \\ \mathrm{K}({\mathrm{D}}_{\mathrm{m}}-{\mathrm{d}}_{\mathrm{i}\mathrm{m}})/{\mathrm{D}}_{\mathrm{m}} \mathrm{T}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} \mathrm{i}\mathrm{s} \mathrm{a} \mathrm{f}\mathrm{l}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{t}& \end{array}\right.$$

(2)

The formula proposed in this paper is Equation (2), which is an original proposal. Here, i represents aircraft node i, m represents the air-route node, dim represents the flight distance of aircraft i on air-route m, and Dm represents the total distance of air-route m.

As can be seen from Formula (2), the greater the proportion of the flight distance of an aircraft on an air-route to the total length of that air-route, the greater the indication that the aircraft is about to fly out of the air-route, and the influence and importance of that aircraft on that air-route will decrease. Therefore, the weight value of that aircraft on that air-route is reduced. If there is a flight conflict during the flight of the aircraft, the weight of the aircraft on the air-route is K times the original value.

The parameter K is used to measure the multiple of the change in the weight of an aircraft on a flight segment when a flight conflict occurs. Its value is determined by comprehensively considering the severity of the flight conflict and the importance of the flight segment, and referring to the aviation industry standards and expert experience. In this case, after comprehensive trade-offs and simulation experiments, the value of K is set to 10. This value can effectively reflect the impact of the conflict, and the model is stable and has good interpretability. Sensitivity analysis verifies that it performs well in different scenarios.

3. Construction of the Interdependent Network in the Guangdong–Hong Kong–Macao Greater Bay Area

This paper selects the actual scenario of air traffic in the multi-airport terminal area of the Guangdong–Hong Kong–Macao region to establish an air-route–flight state interdependent network model for experimental verification. The multi-airport terminal area in the Guangdong–Hong Kong–Macao region consists of three core airports, namely Hong Kong, Guangzhou, and Shenzhen, as well as important airports such as Macao and Zhuhai. There are 148 air-routes in its airspace, featuring high route density and complex relationships.

In the airway chart published by the Civil Aviation Administration of China (CAAC) on 1 August 2023, we identified the names of waypoints in the airspace over the Greater Bay Area, their latitudes and longitudes, and the connection relationships between them. We also collected the air traffic flow data over the Guangdong–Hong Kong–Macao Greater Bay Area at 14:00 on 18 June 2024. The air situation map at this moment is shown in Figure 3. The red dots and numbers represent the operating aircraft and their serial numbers, and the black line segments represent the air-routes. Based on the method described above, we judged the flight conflict situations between aircraft. If there is a flight conflict between two aircraft, there will be a green edge between the corresponding aircraft nodes.

In the interdependent network model, we can use the weighted adjacency matrix of the interdependent network to describe the air traffic situation in the multi-airport terminal area of the Guangdong–Hong Kong–Macao region. Based on the air situation map at the current moment, the weighted adjacency matrix A of the interdependent network model at this moment is calculated, as shown in Figure 4 below.

Based on the adjacency matrix, we can obtain the interdependent network model of the Guangdong–Hong Kong–Macao Greater Bay Area at 14:00 on 18 June 2024, as shown in Figure 5. In the figure, the red dots represent aircraft nodes, the black dots represent air-route nodes, the green line segments are the conflict edges in the flight-state network, the purple line segments are the inter-layer coupling edges, and the connection relationships between air-routes are represented by the blue line segments.

4. Air Traffic Situation Analysis Based on the Interdependent Network

This paper analyzes the traffic situation in the multi-airport terminal area based on the network structure characteristics of the interdependent network. Three network node indicators, namely node degree, weighted clustering coefficient, and node strength, are selected here and integrated using the Analytic Hierarchy Process (AHP) to obtain the overall air traffic situation value.

The node degree reflects the number of air-route nodes that have connection relationships with a given air-route node. It is a key indicator for evaluating the busyness of air-routes and the connectivity of the network, and is of great significance for the analysis of air traffic situations. We use k_i to describe this indicator, and its calculation formula is as follows:

$$k_i={\sum }_{\mathrm{j}=1}^{\mathrm{N}}{\mathrm{a}}_{\mathrm{i}\mathrm{j}}$$

(3)

Let N denote the total number of nodes. Let${ \mathrm{a}}_{\mathrm{i}\mathrm{j} }$represent whether there is an edge between node i and node j. If there is an edge, ${\mathrm{a}}_{\mathrm{i}\mathrm{j}}=1$; otherwise,${\mathrm{a}}_{\mathrm{i}\mathrm{j}}=0$.

The average node degree is the average value of the degrees of all nodes in the network. It is an overall measure of the node degrees of the entire network. The calculation formula for the average node degree is as follows:

$$\bar{k}={\displaystyle \frac{1}{N}}{\sum }_{\mathrm{i}=1}^{N}k\left(\mathrm{i}\right)$$

(4)

Among them, N is the total number of nodes in the network, and (i) is the degree of node i.

The weighted clustering coefficient is an indicator used to measure the local connection tightness around nodes in a complex network, reflecting the tightness of connections between the neighboring nodes around a node and the distribution of edge weights. For a node i, its weighted clustering coefficient C(i) can be calculated by the following formula:

$${\mathrm{c}}_{\mathrm{i}}={\displaystyle \frac{1}{\left({\mathrm{k}}_{\mathrm{i}}-1\right)\times {\mathrm{s}}_{\mathrm{i}}}}\times {\sum }_{\mathrm{m},\mathrm{n}}{\displaystyle \frac{{\mathrm{w}}_{\mathrm{i}\mathrm{m}}+{\mathrm{w}}_{\mathrm{i}\mathrm{n}}}{2}}\times {\mathrm{a}}_{\mathrm{i}\mathrm{m}}{\mathrm{a}}_{\mathrm{i}\mathrm{n}}{\mathrm{a}}_{\mathrm{m}\mathrm{n}}$$

(5)

Let${ \mathrm{k}}_{\mathrm{i} }$represent the node degree$;{ \mathrm{s}}_{\mathrm{i} }$represent the node strength; m and n represent two nodes connected to node i; and${ \mathrm{w}}_{\mathrm{i}\mathrm{m} }$and${ \mathrm{w}}_{\mathrm{i}\mathrm{n} }$ represent the weights of the edges between node i and nodes m and n, and between node i and nodes m and n, respectively.

The average weighted clustering coefficient of nodes is the average value of the weighted clustering coefficients of all the nodes in the network. The calculation formula for the average weighted clustering coefficient of nodes is as follows:

$$\stackrel{\mathrm{‾}}{\mathrm{C}}={\displaystyle \frac{1}{\mathrm{N}}}{\sum }_{\mathrm{i}=1}^{\mathrm{N}}\mathrm{C}\left(\mathrm{i}\right)$$

(6)

Among them, C(i) is the weighted clustering coefficient of node i.

The node strength reflects the busyness of an air-route node. The node strength of a node is the sum of the weights of all its weighted edges. The node strength S(i) of node i is the sum of the weights of the edges between it and all its neighboring nodes, that is

$$s_i={\sum }_{j=1}^N{a}_{ij }{w}_{ij}$$

(7)

Let$\mathrm{N}$represent the total number of nodes and${\mathrm{a}}_{\mathrm{i}\mathrm{j}}$represent whether there is an edge between node i and node j. If there is an edge,${\mathrm{a}}_{\mathrm{i}\mathrm{j}}=1$; otherwise,${\mathrm{a}}_{\mathrm{i}\mathrm{j}}=0$. And${ \mathrm{w}}_{\mathrm{i}\mathrm{j} }$represents the weight of the edge between node i and node j.

The average node strength is the average value of the strengths of all the nodes in the network, which reflects the overall busyness of the entire network. The calculation formula for the average node strength is as follows:

$$\stackrel{\mathrm{‾}}{\mathrm{S}}={\displaystyle \frac{1}{N}}\sum _{\mathrm{i}=1}^N\mathrm{S}\left(\mathrm{i}\right)$$

(8)

Among them, S(i) is the node strength of node i.

During the process of calculating the weight of the average node strength using the Analytic Hierarchy Process (AHP), the differences between the average node strength and the average node degree, and the average weighted clustering coefficient of the nodes, are too large. Therefore, we choose the tanh function to constrain the average node strength within the range of (0, 2).

The overall air traffic situation value is calculated by the weighted combination of the three network situation indicators mentioned above through the Analytic Hierarchy Process (AHP). Its calculation formula is as follows:

$$\mathrm{W}1\times \left(\bar{k}\right)+\mathrm{W}2\times \left(\stackrel{\mathrm{‾}}{\mathrm{C}}\right)+\mathrm{W}3\times \left(1+\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\stackrel{\mathrm{‾}}{\mathrm{S}}\right)\right)$$

(9)

In the formula, W1, W2, and W3 are the indicator weights calculated by the Analytic Hierarchy Process (AHP). These weights reflect the relative importance of the three network node indicators, namely the node degree, weighted clustering coefficient, and node strength, in the overall situation. According to the experimental results,$\mathrm{W}1=0.6370$, W2 = 0.2583, and W3 = 0.1047.

We collected the air situation maps from 14:00 to 15:00 on 18 June to obtain the interdependent network model for this time period. Based on the interdependent network model and the method for calculating the overall air traffic situation value described above, we obtained the overall air traffic situation values at each moment, as shown in

Table 1. Air traffic situation values.

Time Instant	$\stackrel{\mathit{‾}}{\mathit{k}}$	$\stackrel{\mathbf{‾}}{\mathbf{C}}$	$\stackrel{\mathbf{‾}}{\mathbf{S}}$	Situation Value
14:05	7.39	0.62	11.98	14.85
14:10	6.24	1.58	10.84	13.93
. . .	. . .	. . .	. . .	. . .
14:14	7.56	0.94	28.53	17.63
14:16	5.23	0.25	5.27	3.66
. . .	. . .	. . .	. . .	. . .
14:18	6.47	1.66	9.65	14.87
14:36	7.46	1.24	10.92	15.45
. . .	. . .	. . .	. . .	. . .
14:40	8.98	1.85	10.45	16.36
14:44	6.78	0.78	10.36	17.42
. . .	. . .	. . .	. . .	. . .
14:47	10.38	1.89	29.3	26.46
14:53	7.18	1.28	10.52	13.97
. . .	. . .	. . .	. . .	. . .
15:00	5.66	2.37	12.77	17.55

below.

Considering that we collected air situation maps within one hour with a time interval of one minute, theoretically, Table 1 should present the situational values corresponding to 60 moments. However, to present the data more clearly and effectively, we selected some representative situational values, which cover the maximum and minimum situational values, from these 60 moments and included them in Table 1. The three figures in Figure 6 correspond to the air situation maps at the moments when the situational values are at their minimum, normal, and maximum within this hour, respectively, intuitively presenting the air traffic conditions under different situations.

As can be seen from Table 1, the air traffic situation value in the Guangdong–Hong Kong–Macao Greater Bay Area ranges from 14 to 18 within the one-hour period from 14:00 to 15:00, indicating that the overall air traffic situation value should be maintained between 14 and 18 during normal operation. The value is the smallest at 14:16, with a value of 3.6, which means that the number of aircraft flying over the multi-airport terminal area and the number of flight conflicts are relatively small at this moment. The value is the largest at 14:47, with a value of 26.4, indicating that the number of aircraft flying over the multi-airport terminal area and the number of flight conflicts are relatively large at this moment. We listed the air situation maps of the Guangdong–Hong Kong–Macao Greater Bay Area at three moments: 14:16 (low state), 14:40 (normal state), and 14:47 (high state), as shown in Figure 6.

As can be seen from Figure 6, at 14:16, the number of aircraft flying over the terminal area was only 136. sorties The 29 green line segments in the figure indicate that there were 29 pairs of aircraft with flight conflicts. It can be inferred that there were relatively few flight conflicts among the three major core airports of Hong Kong, Guangzhou, and Shenzhen. At 14:40, the number of aircraft operating over the terminal area was 206, and there were 46 pairs of aircraft with flight conflicts. For air traffic controllers, the dispatching pressure was not high. At 14:47, the number of aircraft operating over the terminal area was 308, and there were 92 pairs of aircraft with flight conflicts. At this time, it was the peak period in the multi-airport terminal area of the Guangdong–Hong Kong–Macao region.

This paper collected air situation maps of the Guangdong–Hong Kong–Macao Greater Bay Area from 14:00 to 15:00 on 18 June, with a time interval of one minute. Based on the modeling method described above, a flight segment–flight state interdependent network model was constructed for this time period. The overall air traffic situation value was calculated by comprehensively considering three network indicators: the node degree, weighted clustering coefficient, and node strength. Subsequently, the moments corresponding to the highest, lowest, and normal levels of the air traffic situation value within this period were identified. The number of aircraft conflict pairs from the air situation maps and the number of green conflict edges in the interdependent network model at these three moments, and the actual situations in the air situation maps at these moments, were extracted for verification analysis.

From this, it can be seen that the overall situation value in the interdependent network is consistent with the actual air traffic situation in the multi-airport terminal area of the Guangdong–Hong Kong–Macao region. Therefore, it can be verified that the air-route–flight state interdependent network model can effectively reflect the air traffic situation.

5. Analysis of Air Traffic Situation Evolution

A time series is a set of numerical sequences in which the observed variables in a system are arranged in chronological order, which can reflect the evolution law of a complex system [17]. In this paper, from the perspective of time series, the evolution of the air traffic situation in the multi-airport terminal area is analyzed. The time series is analyzed from the perspectives of the phase-portrait method and chaos. Based on the air situation maps of the Guangdong–Hong Kong–Macao Greater Bay Area from 12:00 to 22:00 on 18 June 2024, the original data’s time series is plotted, with a time interval of 1 min and a total duration of 600 min.

The phase-portrait method transforms a one-dimensional time series into a trajectory graph in a high-dimensional space through embedding to analyze the dynamic characteristics of the sequence. In the phase space, the graph corresponding to chaotic motion is an unclosed curve randomly distributed within a certain area. We selected the time series of the node degree, node strength, and weighted clustering coefficient of the air-route node No. 129, and the overall air traffic situation value for analysis. A two-dimensional phaseportrait of the time series is analytically constructed using a difference equation. The original time series and its two-dimensional phase portrait are shown in Figure 7. This paper converts a one-dimensional time series into a trajectory graph in two-dimensional space.

In this study, the AHP and chaos analysis (including phase diagrams, Lyapunov exponents, and SSA, i.e., Singular Spectrum Analysis) play different but important roles. The AHP divides the complex problem of air traffic situation assessment in the multi-airport terminal area into levels such as objectives, criteria, and alternatives. By constructing a judgment matrix based on expert scores, it determines the weights of each indicator, laying the foundation for subsequent comprehensive assessments. Phase diagrams project high-dimensional air traffic situation data into a two- or three-dimensional space, visually showing the evolution trajectory of the system state over time, which helps to identify the dynamic patterns of the system and possible attractors, so as to understand the overall behavioral characteristics of the system. Lyapunov exponents are used to quantitatively measure the degree of chaos in the system. If the exponent is positive, it indicates that the system is sensitive to initial conditions and has chaotic behavior, which is helpful for understanding the unpredictability and complexity of the air traffic situation. SSA decomposes and reconstructs the original situational time-series data, dividing it into trend terms, periodic terms, and noise terms. By analyzing each component, it extracts meaningful information such as potential periodic patterns and abnormal fluctuations, providing a basis for the prediction and management of the air traffic situation.

The calculation formula for the second-order central difference value is as follows:

$$f_{\left(x_0\right)}^{\prime }={\displaystyle \frac{f_{\left(x_0+t\right)}-f_{\left(x_0-t\right)}}{2t}}$$

(10)

In the formula t is the time interval; $f_{\left(x_0+t\right)}$and ${ f}_{\left(x_0-t\right)}$are the values of$f_{\left(x_0\right)}$at the adjacent previous and subsequent moments, respectively.

As can be seen from Figure 8, although the two-dimensional phase portraits of the time series of node characteristics in the interdependent network model are rather chaotic, they still exhibit certain periodic trajectories. Chaos is a complex nonlinear dynamical phenomenon, characterized by a seemingly random and disordered motion form of the system during the deterministic evolution process, but actually there are inherent laws. From the perspective of chaos, we verified whether the time series of the indicators of this interdependent network model are predictable.

We verified the chaos of the system based on the maximum Lyapunov exponent [18]. When the maximum Lyapunov exponent is greater than 0, the system can be considered to have chaotic characteristics. The results are shown in Figure 8.

From Figure 8, it can be observed that the maximum Lyapunov exponents corresponding to the time series of air-route node characteristics are all greater than or equal to 0. This result indicates that the time series of node characteristics exhibits chaotic features. The chaos of the system means that the changes in the evolution law of the air traffic system are predictable to a certain extent. It is worth noting that the maximum Lyapunov exponents of some nodes are 0. This is because these nodes are in an isolated state during the evolution process, with relatively stable states, and are not significantly affected by external factors or have weak interactions with other nodes.

As can be seen from Figure 7, the original time series of node characteristics can accurately reflect the instantaneous changes in node characteristics, but it lacks smoothness as a whole and has many obvious turning points, which makes it difficult to judge its changing trend. The corresponding two-dimensional phase portrait shows a certain regularity, but the change pattern of the curve is not obvious, and the overall graph appears rather chaotic with low smoothness. The main reason for these problems is that the node characteristic data are calculated from the actual air situation and are affected by the internal chaos of the system. To solve this problem, we used the Singular Spectrum Analysis method [19] to smooth the original data and obtain the smoothed-node characteristic sequences and two-dimensional phase portraits, as shown in Figure 9.

As can be seen from Figure 9, after the smoothing process, the time series, and the two-dimensional phase portraits of node characteristics and are clearer and more concise overall, which facilitates observation and analysis. The curves of the time series of node characteristics change more smoothly and tend to be stable, eliminating the obvious turning points that previously existed. For the two-dimensional phase portraits after smoothing, the degree of graph chaos is reduced, the interference among the graphs of different nodes is decreased, and the regularity is more obvious. In the smoothed two-dimensional phase portraits, the graphs of different nodes have geometric self-similarity, indicating that the system is not random during the evolution process but has obvious chaos. Therefore, it is feasible to predict its evolution process later.

6. Conclusions

This paper proposes a method for analyzing the evolution of the air traffic situation in the multi-airport terminal area: an air-route–flight state interdependent network model is constructed, where the aircraft network and the air-route network are connected through coupling edges.

Based on three indicators, namely the node degree, node strength, and weighted clustering coefficient, the overall situation during the evolution process is analyzed. Verification shows that this method can accurately reflect the air traffic situation. Through the analysis of the maximum Lyapunov exponents and the two-dimensional phase portraits of the time series of the key indicators of the interdependent network and the overall air traffic situation, it is confirmed that the situation evolution in the multi-airport terminal area has chaos characteristics.

In the future, situation assessment and prediction can be carried out based on this research to provide accurate status information for air traffic control.