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Abstract

An urban public transportation super-network model is established in this article. Compared with three existing complicated urban public transportation network models (the C-space network, L-space network, and P-space network), our urban public transportation super-network model first incorporated both station networks and bus line networks explicitly, which connects two non-homogeneous nodes (station and line nodes) through the notion of super networks. The proposed model reflects the interaction and mutual influences between the internal features and internal structures of the urban public transportation super network. And then the function projective synchronization of the dynamic public transportation super-network model is proven, and the high stability of the public transportation super network is confirmed by the Lyapunov stability theorem. Finally, the classical Lorenz chaotic system is applied in a numerical example to illustrate the proposed model and algorithm. The impact of departure frequency, line crowdedness, passenger transport density, and station dwell time on synchronous ability of the urban public transportation super-network model is analyzed. So, this study is a discussion on the urban public transportation super-network balance in order to offer theoretical basis for people to research bus dispatching and bus lines optimization.

Keywords

Super network urban public transportation function projective synchronization time-varying delay

Introduction

The most complicated systems can be described by networks; a complicated network is typically used to describe complicated systems. Common complicated networks comprise many nodes and connecting lines. These nodes are homogeneous and involve connecting lines that represent connections between nodes. Areal complicated network includes connections between connecting lines and between nodes and connecting lines apart from the connections between nodes. A common complicated network cannot fully depict the internal features of a real network. For example, one method to describe an urban public transportation network that adopts a complicated network uses bus lines as nodes. If two bus lines involve a common station, then these two nodes are connected. This method reflects the relationship of transferring between two bus lines. However, one bus line covers many stations. Several stations comprise a neighborhood, whereas several do not comprise a neighborhood but can be reached through transfer. Therefore, relations between nodes as well as between nodes and bus lines cannot be simultaneously described in this network. In the past, problems between two systems of different natures could be solved by bipartite graph, which expresses stations with a group of independent node sets and bus lines with another group of independent node sets. Nevertheless, these two groups of node sets hold dissimilar values and lose node homogeneity. That is, the nodes between these two groups cannot be connected and thus lack information. These problems cannot be solved by bipartite graph, and multi-part graphs can be addressed by a super graph. The graph considers individuals as nodes, and related nodes are connected through a line; one line that covers several nodes is called a super graph. Berge^1,2 proposed the theoretical concept and basic properties of a super graph in 1970. Nagurney and Dong,³ an American scientist, called networks with attributes beyond those of existing networks, “super networks,” when processing such interlaced networks. This description provided an explicit definition to the super network. Academician Z-T Wang and Professor Z-P⁴ Wang further suggested the features of a super network. Existing research on super networks is mainly based on variational inequality.^5,6 They mentioned studies have mainly converted hierarchical and multi-standard super-network equilibrium models into an optimization problem that was solved using evolved variation inequality. Studies based on science system^7–11 investigated the super network from global and local perspectives. These works include studies on the relationship between networks, the relationship between the outside world and the network, and on the overall performance of super networks.

In recent years, with the emergence of super-network research, the application of super network in all fields has been given much attention, such as electricity network,¹² social network,^13,14 the knowledge structure network,¹⁵ and supply chain network. In the field of transportation, the super-network research mainly focused on the optimization study of the equilibrium model of transportation supply chain network; Xu and Gao¹⁶ proposed a new and efficient algorithm for solving supply chain network equilibrium and equivalent super network–based traffic network equilibrium. The new method was called semi-smooth Levenberg–Marquardt-type method which was verified more efficient than the Quasi Newton method and the modified projection method to solve the problem. Kang and Kurtz¹⁷ presented a discrete network design problem for optimally designing freight transport network in terms of the efficiency of supply chain. Modeling was undertaken within the framework of mathematical programs, with equilibrium constraints, which first incorporated both supply chain and transport networks explicitly. Using urban traffic system as a giant complex system, the method of super network has provided a new perspective for researching its complexity and internal dynamic structure. B Si et al.,¹⁸ proposed a super-network model to describe such a multi-mode urban transport system. By considering analysis of travelers’ combined choices and the interferences between different modes on the same road segments, they presented a variation inequality model to describe equilibrium assignment for multi-mode urban transport system. Nevertheless, the research on super network in the urban traffic system is still little at home and abroad.

This article combined the super network concept and the traditional modeling method for complicated urban public transportation networks and established an urban public transportation super-network model. This generated super-network model fully considers the relationships between stations, between stations and lines, and between lines. Unlike the traditional complicated public transportation network model, the proposed super-network model depicts the internal structural characteristics of public transportation networks. The new model describes the relationships between lines and between stations as well as the relationships between stations and lines. Both stations and lines are nodes in this super-network model. Line nodes with transferrable common stations are connected, and all lines belong to a line set $(l_{i} \in L, i = 1, 2, 3)$ . Station nodes directly accessible by bus are connected, and all stations belong to the station set $(s_{j} \in S, j = A, B, C)$ . If one node in the station set belongs to one node in the line set, then these two nodes are connected. The proposed model reflects the interaction and mutual influences between the internal features and internal structures of urban public transportation networks in actual settings. Therefore, the super-network-based dynamic model of urban public transportation networks can more effectively reflect the internal dynamic evolution of the public transportation network and is more useful than the traditional model. In this article, the function projective synchronization of the public transportation super network is studied by the drive–response function projective synchronization method and the Lorenz chaotic system. Finally, the effects of departure frequency, passenger transport density, line crowdedness, and station dwell time on the stability of the super network are numerically analyzed. The results provide theoretical foundations for optimizing public transportation networks and bus dispatching.

Establishment of the urban public transportation super-network model

Three types of complicated urban public transportation networks exist, namely, C-space, P-space, and L-space networks. The first two types regard stations as nodes, whereas the third type uses lines as nodes. The nodes in these three networks are homogeneous. Specifically, only a station node set or line node set is present in one network. This article establishes an urban public transportation super network based on the C- and L-space networks. The method proposed in this article utilizes both stations and lines in the urban public transportation network as nodes. All line nodes belong to the line set $(l_{i} \in L, i = 1, 2, 3)$ , and all the station nodes belong to the station set $(s_{j} \in S, j = A, B, C)$ . In $(l_{i} \in L, i = 1, 2, 3)$ , two nodes with a common station are connected. In $(s_{j} \in S, j = A, B, C)$ , two nodes that are adjacent stations of the same line are connected. The network comprising $(l_{i} \in L, i = 1, 2, 3)$ and connection lines is called the L–L sub-network, whereas the network comprising $(s_{j} \in S, j = A, B, C)$ and connection lines is called the S–S sub-network. If one station in the S–S sub-network belongs to one line in the L–L sub-network, then these station and line nodes are connected.

L–L sub-network model

Lines are obtained as nodes. If two lines hold a common station, then they are connected. In this regard, a line-to-line network called an “L–L sub-network” can be established as follows

G_{L} = (L, E_{L - L})

(1)

where $L = {L_{1}, L_{2}, \dots, L_{n}}$ is the line set, $E_{L - L} = {(L_{i}, L_{j}) | θ (L_{i}, L_{j}) = 1}$ is the connection line set, and $θ (L_{i}, L_{j}) = 1$ indicates direct transfer between lines i and j.

S–S sub-network model

The S–S sub-network takes stations as nodes. If two adjacent stations have the same line, then they are connected. In this way, a station-to-station network is established and called “S–S sub-network” as follows

G_{P} = (S, E_{P - P})

(2)

where $S = {S_{1}, S_{2}, \dots, S_{k}}$ is the station set, $E_{S - S} = {(S_{i}, S_{j}) | λ (S_{i}, S_{j}) = 1}$ is the connection line set, and $λ (S_{i}, S_{j}) = 1$ is the correlation between nodes i and j.

Urban public transportation super-network model

The mapping relationships between the L–L and S–S sub-networks are listed below:

1. Mapping from an L–L sub-network to an S–S sub-network: This relationship reflects the stations on one bus line. We call such super network as a “line–station super network (L(S)SN).”

If station $s_{k}$ belongs to line $l_{i}$ , $ϕ (s_{k}, l_{i}) = 1$ ; otherwise, $ϕ (s_{k}, l_{i}) = 0$ . Therefore, the set of stations in S, which are related with $l_{i}$ , is obtained as follows

S (l_{i}) = f (l_{i}, S) = {s_{k} | s_{k} \in S, ϕ (s_{k}, l_{i}) = 1}

(3)

2. Mapping from an S–S sub-network to an L–L sub-network: This relationship reflects the bus lines that cover the same station. We call such super network as “station–line super network (S(L)SN).”

Correspondingly, the set of lines in L related to $s_{k}$ is as follows

L (s_{k}) = φ (l_{i}, S) = {l_{i} | l_{i} \in L, ϕ (s_{k}, l_{i}) = 1}

(4)

where $S (l_{i})$ is the set of stations that belong to line $l_{i}$ and $L (s_{k})$ is the set of lines passing station $s_{k}$ .

L(S)SN and S(L)SN are dual networks. The above-mentioned mapping relationships suggest that two urban public transportation super networks can be established. A public transportation super network that involves three lines and nine stations is shown in Figure 1. The line layer and station layer are topologies of L(S)SN and S(L)SN, respectively. The super-network relation is given by the following equation

USN = (G_{L}, G_{S}, E_{L - S}) = (L, S, E_{L - L}, E_{S - S}, E_{L - S})

(5)

Figure 1.

Topology map of public transport super networks.

Conversion of public transportation super-network model

The nodes in the S–S and L–L sub-networks are non-homogeneous and shall thus be treated differently. Therefore, we convert proposed urban public transportation super network that involves these two node types. According to the definition and division of super-network factions, one line node and all stations on this line in L(S)SN can be viewed as one faction (Figure 2). Hence, the proposed super network can be divided into three factions (Figure 3(a)). Obviously, all stations in every faction belong to corresponding line nodes, and the common station of two lines can be expressed by a virtual station on the other line. In this way, L(S)SN can be converted into a new public transportation super network (Figure 3(b)) called new super network (NSN).

Figure 2.

Division of line factions in the super network.

Figure 3.

New public transportation super network: (a) line network and (b) conversion of super network.

Description of the urban public transportation super-network model

NSN is a nested network with both stations and lines (Figure 3). The station network is the inner network, whereas the line network is the outer network. Therefore, this article employs the drive–response-complicated dynamic network model^12,13 to study the projective synchronization of NSN. If the station network is viewed as the drive network, then the line network is the response network.

The following equation shows a network that involves time-varying delays and comprises sub-networks with N same nodes

\begin{array}{l} {\dot{x}}_{i} = C x_{i} (t) + f (x_{i} (t), t) + \sum_{j = 1}^{N} ε_{1} a_{i j} \cdot Γ x_{j} (t) \\ + \sum_{j = 1}^{N} ε_{2} a_{i j} \cdot Γ x_{j} (t - τ), i = 1, 2, …, N \end{array}

(6)

where ${\overset{\cdot}{x}}_{i} (t) = (x_{i 1} (t), x_{i 2} (t), \dots, x_{in} (t))^{T} \in R^{n}$ is the state variable of the ith node at time t. $C \in R^{n \times n}$ is a constant matrix and $C x_{i}$ is the matrix function that describes the original dynamic behavior of each node. $f (x_{i} (t), t) : R^{n} \to R^{n}$ is a smooth nonlinear function and a vector function that describes the original dynamic behavior of each node. $Γ \in R^{n \times n}$ is the internal coupling function between nodes in the network.

Matrix $A = (a_{ij})_{N \times N} \in R^{n \times n}$ is the topological structure of the super network meeting the dissipation coupling condition $\sum_{j = 1}^{N} a_{ij} = 0$ . If the same route passes through two nodes, the $a_{ij} = a_{ji} = 1, (i \neq j)$ links that connect these two nodes exist. Otherwise, $a_{ij} = a_{ji} = 0, (i \neq j)$ . The diagonal element of matrix A is

a_{ii} = - \sum_{j = 1, j \neq i}^{N} a_{ij} = - \sum_{j = 1, j \neq i}^{N} a_{ji}, (i = 1, 2, \dots, N)

Constant $ε > 0$ is the coupling strength and $τ$ is the coupling delay.

If system (5) is regarded as the drive network, the response network is as follows

\begin{array}{l} {\dot{y}}_{i} = C y_{i} (t) + f (y_{i} (t), t) + \sum_{j = 1}^{N} ε_{1} a_{i j} \cdot Γ y_{j} (t) \\ + \sum_{j = 1}^{N} ε_{2} a_{i j} \cdot Γ y_{j} (t - τ) + u_{i} (t), i = 1, 2, …, N \end{array}

(7)

where $u_{i}$ is the input control.

Stability analysis of the urban public transportation super-network model

Definition 1

The complex network system (5) achieves projective synchronization if it satisfies the following relation

lim_{t \to \infty} ‖ x_{i} (t) - α_{i} s (t) ‖ = 0 (i = 1, 2, \dots, N)

where $α_{i}$ is a nonzero constant. Herein, the synchronous state $s (t) \in R^{n}$ is a single isolated solution, that is, $\overset{\cdot}{s} (t) = f (s (t))$ .

Definition 2

Let $e_{i} (t) = y_{i} (t) - α_{i} x_{i} (t), i = 1, 2, \dots, N$ be the error between the drive network (5) and the response network (6). If $t \to \infty$ and $e_{i} (t) \to 0$ , then the drive network (5) and the response network (6) reach function projective synchronization.

Given Definition 2, the error system equation is written as follows

\begin{matrix} {\overset{\cdot}{e}}_{i} (t) = {\overset{\cdot}{y}}_{i} (t) - {\overset{\cdot}{α}}_{i} (t) x_{i} (t) - α_{i} (t) {\overset{\cdot}{x}}_{i} (t) \\ = C e_{i} (t) + \bar{f} (e_{i} (t), t) + {\bar{h}}_{i} (t) \\ + {\bar{g}}_{i} (t - τ) + u_{i} - {\overset{\cdot}{α}}_{i} (t) x_{i} (t) \end{matrix}

(8)

where

\begin{matrix} \bar{f} (e_{i} (t), t) = f (x_{i} (t), t) - f (y_{i} (t), t) \\ {\bar{h}}_{i} (t) = ε_{1} \sum_{j = 1}^{N} a_{ij} Γ x_{j} (t) - ε_{1} \sum_{j = 1}^{N} a_{ij} Γ y_{j} (t) \\ {\bar{g}}_{i} (t - τ) = ε_{2} \sum_{j = 1}^{N} a_{ij} Γ x_{j} (t - τ) - ε_{2} \sum_{j = 1}^{N} a_{ij} Γ y_{j} (t - τ) \end{matrix}

Lemma

Any $x, y \in R^{n}$ and positive definite matrix $Q \in R^{n \times n}$ , $2 x^{T} y \leq x^{T} Qx + y^{T} Q^{- 1} y$ is true.

Hypothesis 1

If a non-negative constant $λ_{ij}, γ_{ij}, (1 \leq j \leq N)$ exists, then $1 \leq i \leq N$ is met

‖ {\bar{h}}_{i} (t) ‖_{2} \leq \sum_{j = 1}^{N} λ_{ij} {‖ e_{j} ‖}_{2}

‖ {\bar{g}}_{i} (t - τ) ‖_{2} \leq \sum_{j = 1}^{N} γ_{ij} {‖ e_{j} (t - τ) ‖}_{2}

Theorem 1

If Hypothesis 1 is true, then the following adaptive controller and parameter updating rules are used

u_{i} (t) = - f (e_{i} (t), t) - d_{i} e_{i} (t) + \overset{\cdot}{α} (t) x_{i} (t)

(9)

{\overset{\cdot}{d}}_{i} = k_{i} {e_{i}}^{T} (t) e_{i} (t) = k_{i} ‖ e_{j} (t) ‖_{2}^{2}

(10)

Then, the drive network (5) and the response network (6) reach function projective synchronization. $k_{i}$ is the time-varying feedback gain, and $l_{i}$ and $r_{i}$ are arbitrary positive constants where $i = 1, 2, \dots, N$ .

Proof

The Lyapunov stability equation is constructed as follows

V (t) = \frac{1}{2} \sum_{i = 1}^{N} e_{i}^{T} (t) e_{i} (t) + \frac{1}{2} \sum_{i = 1}^{N} \frac{1}{k_{i}} {(d_{i} - d^{*})}^{2}

(11)

where $d^{*}$ is a constant to be determined.

Equation (11) is then derived, and equations (9) and (10) are substituted into equation (11) as follows

\begin{matrix} \overset{\cdot}{V} = \sum_{i = 1}^{N} e_{i}^{T} (t) [C e_{i} (t) + \bar{f} (e_{i} (t), t) + {\bar{h}}_{i} (t) + {\bar{g}}_{i} (t - τ) + u_{i} - {\overset{\cdot}{α}}_{i} (t) x_{i} (t)] + \sum_{i = 1}^{N} \frac{1}{k_{i}} (d_{i} - d^{*}) {\overset{\cdot}{d}}_{i} \\ = \sum_{i = 1}^{N} {Ce}_{i}^{T} (t) e_{i} (t) + \sum_{i = 1}^{N} {e_{i}}^{T} (t) {\bar{h}}_{i} (t) + \sum_{i = 1}^{N} e_{i}^{T} (t) {\bar{g}}_{i} (t - τ) - \sum_{i = 1}^{N} d^{*} e_{i}^{T} (t) e_{i} (t) \\ \leq \frac{1}{2} \sum_{i = 1}^{N} ‖ e_{i} (t) ‖_{2}^{2} (C + C^{T}) + \sum_{i = 1}^{N} \sum_{j = 1}^{N} λ_{ij} {‖ e_{i} (t) ‖}_{2} {‖ e_{j} (t) ‖}_{2} + \sum_{i = 1}^{N} \sum_{j = 1}^{N} γ_{ij} {‖ e_{i} (t) ‖}_{2} {‖ e_{j} (t) ‖}_{2} - \sum_{i = 1}^{N} d^{*} ‖ e_{i} (t) ‖_{2}^{2} \\ = e^{T} (t) [\frac{1}{2} (C + C^{T}) + Λ + Υ + \frac{1}{2 (1 - ε)} - d^{*}] e (t) = e^{T} (t) [θ (t) + Λ + Υ - d^{*}] e (t) \end{matrix}

Let $θ (t) \overset{Δ}{=} (1 / 2) (C + C^{T})$ , $d^{*} = θ (t) + Λ + Υ + ω$ , and $ω > 0$

Then

\overset{\cdot}{V} \leq - ω e^{T} (t) e (t)

(12)

According to the Lyapunov theorem, equation (12) is negatively definite. In other words, the error system (8) achieves global asymptotic stability under equations (9) and (10).

Numerical simulation

We conducted a dynamic simulation of the proposed urban public transportation super network. $N = 3$ , $τ = 0.3$ , and the average coupling strength of node is set as $ε_{1} = ε_{2} = 0.3$ . The following are the internal and external coupling matrices

A = [\begin{matrix} - 2 & 1 & 1 \\ 1 & - 2 & 1 \\ 1 & 1 & - 2 \end{matrix}], Γ = [\begin{matrix} - 2 & 1 & 1 \\ 1 & - 5 & 4 \\ 1 & 4 & - 5 \end{matrix}]

Node changes in this super network meet nonlinear characteristics. Suppose they meet a nonlinear Lorenz system, and let the Lorenz chaotic system be the kinetic equation of nodes. The controller and parameter updating rule are chosen as follows

\begin{matrix} u_{i} (t) = - f (e_{i} (t), t) - d_{i} e_{i} (t) + \overset{\cdot}{α} (t) x_{i} (t) \\ {\overset{\cdot}{d}}_{i} = k_{i} {e_{i}}^{T} (t) e_{i} (t) = k_{i} ‖ e_{j} (t) ‖_{2}^{2}, i = 1, 2, 3 \end{matrix}

When i = 1

\begin{array}{l} (\begin{matrix} {\dot{x}}_{11} (t) \\ {\dot{x}}_{12} (t) \\ {\dot{x}}_{13} (t) \end{matrix}) = (\begin{matrix} - 10 x_{11} (t) + 10 x_{12} (t) \\ 28 x_{11} (t) - x_{12} (t) - x_{11} (t) x_{13} (t) \\ - \frac{8}{3} x_{13} (t) + x_{11} (t) x_{12} (t) \end{matrix}) \\ + ε_{1} A \cdot Γ x_{j} (t) + ε_{2} A \cdot Γ x_{j} (t - τ) + μ_{1} \end{array}

where

A_{1} (t - τ) = (\begin{matrix} a_{11} x_{11} (t - τ) + a_{12} x_{21} (t - τ) + a_{13} x_{31} (t - τ) \\ a_{11} x_{12} (t - τ) + a_{12} x_{22} (t - τ) + a_{13} x_{32} (t - τ) \\ a_{11} x_{13} (t - τ) + a_{12} x_{23} (t - τ) + a_{13} x_{33} (t - τ) \end{matrix})

When i = 2

\begin{array}{l} (\begin{matrix} {\dot{x}}_{21} (t) \\ {\dot{x}}_{22} (t) \\ {\dot{x}}_{23} (t) \end{matrix}) = (\begin{matrix} - 10 x_{21} (t) + 10 x_{22} (t) \\ 28 x_{21} (t) - x_{22} (t) - x_{21} (t) x_{23} (t) \\ - \frac{8}{3} x_{23} (t) + x_{21} (t) x_{22} (t) \end{matrix}) \\ + ε_{1} A \cdot Γ x_{j} (t) + ε_{2} A \cdot Γ x_{j} (t - τ) + μ_{2} \end{array}

where

A_{2} (t - τ) = (\begin{matrix} a_{21} x_{11} (t - τ) + a_{22} x_{21} (t - τ) + a_{23} x_{31} (t - τ) \\ a_{21} x_{12} (t - τ) + a_{22} x_{22} (t - τ) + a_{23} x_{32} (t - τ) \\ a_{21} x_{13} (t - τ) + a_{22} x_{23} (t - τ) + a_{23} x_{33} (t - τ) \end{matrix})

When $i = 3$

\begin{array}{l} (\begin{matrix} {\dot{x}}_{31} (t) \\ {\dot{x}}_{32} (t) \\ {\dot{x}}_{33} (t) \end{matrix}) = (\begin{matrix} - 10 x_{31} (t) + 10 x_{32} (t) \\ 28 x_{31} (t) - x_{32} (t) - x_{31} (t) x_{33} (t) \\ - \frac{8}{3} x_{33} (t) + x_{31} (t) x_{32} (t) \end{matrix}) \\ + ε_{1} A \cdot Γ x_{j} (t) + ε_{2} A \cdot Γ x_{j} (t - τ) + μ_{3} \end{array}

where

A_{3} (t - τ) = (\begin{matrix} a_{31} x_{11} (t - τ) + a_{32} x_{21} (t - τ) + a_{33} x_{31} (t - τ) \\ a_{31} x_{12} (t - τ) + a_{32} x_{22} (t - τ) + a_{33} x_{32} (t - τ) \\ a_{31} x_{13} (t - τ) + a_{32} x_{23} (t - τ) + a_{33} x_{33} (t - τ) \end{matrix})

The numerical simulation is accomplished with MATLAB software7.0. Results are shown in Figure 4, where the x-axis corresponds to time and the y-axis corresponds to the coupling error between the drive network and the response network. If error = 0, then the super network tends to be stable.

Figure 4.

Synchronization errors of the public transport super networks when $ε_{1} = ε_{2} = 0.3, τ = 0.3$ .

In this article, network stability refers to the dynamic equilibrium between passengers in the queue at stations, the degree of crowdedness, and the departure frequency in the urban public transportation super network. All passengers waiting in every station can board a bus in the shortest possible time. When the lag between lines is 0.3 and the coupling strength of the nodes is 0.3, at least six unit times are required for the super network to stabilize.

This article explores the effects of departure frequency, line crowdedness, passenger transport density, and station dwell time on super network stability. The simulation results under $ε_{1} = ε_{2} = 0.5, τ = 0.1$ are displayed in Figure 5.

Figure 5.

Synchronization errors of the public transport super networks when $ε_{1} = ε_{2} = 0.5, τ = 0.1$ .

The time for the network to achieve stability is shortened by two unit times (Figure 5). This finding implies that higher departure frequencies and shorter station dwell times shorten the time for the super network to achieve stability. Therefore, passenger waiting time and heavy passenger stranding can be relieved effectively by adjusting vehicle scheduling, reducing station dwell time during passenger-flow rush hours or in large junction stations, and increasing bus quantity.

Conclusion

This article establishes an urban public transportation super-network model. Unlike the traditional public transportation network model, the proposed super-network model depicts two non-homogeneous station nodes and line nodes simultaneously through reasonable connections. This strategy more effectively conforms to the practical public transportation network and more completely reflects the internal structural characteristics of nodes and lines in the network than the traditional model. Next, the proposed super network is converted into a nested public transportation super network based on the division theory of super-network factions, which will aid in future study. The stabilities of the drive network (S–S sub-network) and response network (L–L sub-network) are validated by the classical Lyapunov stability theorem. Furthermore, sufficient conditions for function projective synchronization between the drive–response networks under time-varying delays are obtained. Finally, the stability of the proposed super network is confirmed based on the Lorenz chaotic system. The effects of departure frequency, line crowdedness, passenger transport density, and station dwell time on super network stability are analyzed.

Footnotes

Acknowledgements

The authors thank the referees and the editor for their comments and suggestions.

Academic Editor: Hai Xiang Lin

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (Nos 51408288, 61563029, and 61164003).

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Analysis of the projective synchronization of the urban public transportation super network

Abstract

Keywords

Introduction

Establishment of the urban public transportation super-network model

L–L sub-network model

S–S sub-network model

Urban public transportation super-network model

Conversion of public transportation super-network model

Description of the urban public transportation super-network model

Stability analysis of the urban public transportation super-network model

Definition 1

Definition 2

Lemma

Hypothesis 1

Theorem 1

Proof

Numerical simulation

Conclusion

Footnotes

Acknowledgements

Declaration of conflicting interests

Funding

References