A traffic flow congestion intervention method based on saddle-node bifurcation control

Abstract

The present application belongs to the technical field of intelligent traffic control, and particularly relates to a traffic flow congestion intervention method based on saddle-node bifurcation control, comprising the following steps: S1, obtaining a macroscopic traffic flow dynamics equation describing the spatiotemporal evolution of traffic density and macroscopic speed based on a full speed difference model; S2, introducing a traveling wave transformation z=x-ct, converting it into a second-order ordinary differential equation about traffic density, and further expressing it as a first-order ordinary differential dynamic system taking density and its derivative with respect to the traveling wave coordinate z as state variables; S3, constructing a nonlinear feedback control signal; and S4, introducing the nonlinear feedback control signal as a control item into the first-order ordinary differential dynamic system of S2 to form a controlled dynamic system. The present method can achieve source inhibition before congestion is formed.

Description

Technical Field

[0001] This invention belongs to the field of intelligent traffic management technology, and in particular relates to a traffic flow congestion intervention method based on saddle node bifurcation control. Background Technology

[0002] With the acceleration of global urbanization and the continuous increase in motor vehicle ownership, traffic congestion has become a key bottleneck restricting sustainable urban development, reducing economic efficiency, and exacerbating environmental pollution. Building an efficient, safe, and intelligent modern transportation system urgently requires a deep understanding of the nonlinear dynamic evolution mechanism of traffic flow and, based on this, the development of forward-looking and proactive congestion intervention strategies. Traffic flow is essentially a typical nonlinear dissipative dynamic system, characterized by the potential for abrupt instability under specific traffic density or speed conditions—a transition from a free-flowing, stable state to a congested state, accompanied by the emergence of complex spatiotemporal structures such as periodic stop-and-go waves and localized congestion clusters. Such phase transitions often stem from the coupling effect of internal nonlinear feedback and external disturbances, exhibiting high sensitivity and irreversibility, thus posing a severe challenge to the stability analysis and proactive control of transportation systems.

[0003] At the theoretical modeling level, micro-level car-following models and macro-level continuous models constitute the two pillars of traffic flow dynamics research. Micro-level models focus on the interactions between individual vehicles (such as acceleration response, reaction time delay, and field of vision). In recent years, research on "multiple preceding vehicle influences" has significantly deepened our understanding of the complexity of driving behavior: from early optimized speed (OV) models to full velocity difference (FVD) models that incorporate speed difference feedback; and then to improved car-following models proposed by D. Ngoduy, Hua, Peng, and others, which integrate multiple preceding vehicle expectations, average speed effects, and even following vehicle feedback, all these developments demonstrate that driver decisions not only depend on the vehicle immediately in front but are also influenced by the coordinated movement of several vehicles ahead. Such models exhibit stronger congestion mitigation potential in vehicle-to-vehicle (V2V / V2I) environments. Simultaneously, macro-level modeling has also evolved: scholars such as Zhai, Hu, and Ngoduy have embedded the average speed of multiple preceding vehicles, bidirectional perception, and predictive mechanisms into continuous equations, effectively improving the model's prediction accuracy and robustness under complex road networks and geometric constraints.

[0004] To improve system stability, researchers have further incorporated control theory into traffic flow modeling frameworks. Tang et al. introduced real-time traffic information feedback to enhance the adaptability of macroscopic models; Cheng et al., based on intelligent feedback and predictive control, designed dynamic controllers in networked and mixed traffic scenarios, achieving congestion relief and traffic efficiency optimization by adjusting vehicle acceleration or signal timing. While these works have made progress in improving system robustness, their control logic largely relies on empirical thresholds or linear approximations, lacking in-depth analysis of the inherent nonlinear instability mechanisms of traffic flow, and making it difficult to accurately capture subtle signs before critical instability.

[0005] Bifurcation theory in nonlinear dynamics provides a key analytical tool for such problems. Saddle-node bifurcation, as a typical local bifurcation, characterizes the critical process of the pairing of equilibrium points in a system, directly corresponding to the disappearance of the boundary between the "free flow" and "congested flow" coexisting states in traffic flow. That is, when the traffic density exceeds a certain critical value, the uniform flow solution suddenly loses its existence, and the system is forced to transition to a high-density congested state. Jin et al. have confirmed that saddle-node and Hopf bifurcations can occur in the FVD model under parameter variations, inducing density abrupt changes and oscillatory waves, respectively. Habib and Marcial et al. have explored model-free transient identification and deep learning-driven saddle-node localization methods, respectively, expanding their application boundaries in fields such as power systems.

[0006] However, although saddle-knot bifurcation can theoretically accurately characterize the critical conditions for traffic loss stability, existing research still has obvious limitations: (1) Disconnect between analysis and control: Most literature only stays at the stage of identifying bifurcation phenomena and explaining mechanisms, and rarely transforms the critical conditions of saddle-knot bifurcation into an operational basis for active intervention; (2) Static control strategies: Existing feedback control is mostly based on fixed thresholds or linear feedback laws, such as proportional-integral control, without considering the dynamic drift characteristics of bifurcation points with traffic conditions, resulting in control lag or over-adjustment; (3) Insufficient nonlinear coupling modeling: Current macroscopic models often ignore high-order nonlinear terms induced by microscopic multi-vehicle interactions, such as density square terms, resulting in large deviations in the calculation of bifurcation critical points, which weakens the physical credibility of control strategies; (4) Lack of real-time adaptability: The actual traffic environment has strong time-varying and spatial heterogeneity, while existing bifurcation analysis is mostly based on steady-state assumptions or idealized boundary conditions, which is difficult to support the needs of online and distributed intervention.

[0007] Crucially, saddle-node bifurcation is inherently irreversible and has a hysteresis effect—once the system crosses the bifurcation point, even if parameters are adjusted in the opposite direction, the original stable branch cannot be immediately restored. This means that traditional passive congestion mitigation measures, such as flow restriction and detour guidance, often only take effect after the bifurcation has occurred, by which time congestion has already formed and begun to spread, significantly increasing the cost of mitigation.

[0008] Therefore, how to curb congestion at its source before it occurs has become an urgent problem to be solved. Summary of the Invention

[0009] To address the aforementioned shortcomings of existing technologies, the present invention aims to provide a traffic flow congestion intervention method based on saddle node bifurcation control, which can achieve source suppression before congestion occurs.

[0010] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is as follows:

[0011] A traffic flow congestion intervention method based on saddle node bifurcation control includes the following steps:

[0012] S1. Based on the full speed difference model, the perception of speed differences between multiple vehicles ahead is introduced to construct a micro-car-following model. Through macroscopic continuous derivation of the micro-car-following model, a model describing traffic density is obtained, with spatial location x and time t as independent variables. and macro speed Macroscopic traffic flow dynamics equations of spatiotemporal evolution;

[0013] S2. Introduce a traveling wave transformation z = x - ct into the macroscopic traffic flow dynamics equation of S1, where c is the wave velocity, to transform it into a function relating to traffic density. The second-order ordinary differential equation, further expressed as density and its derivative with respect to the traveling wave coordinate z A first-order ordinary differential dynamical system with state variables; determine the equilibrium point of the first-order ordinary differential dynamical system. ,in The traffic density of the system under uniform steady state is calculated, and its Jacobian matrix is ​​calculated at the equilibrium point. Based on the saddle node bifurcation criterion that the determinant of the Jacobian matrix is ​​zero, the critical condition for the system to undergo saddle node bifurcation is determined.

[0014] S3. Construct a nonlinear feedback control signal Where x is the traffic density relative to the equilibrium point density; The disturbance quantity; y is the state variable in S2. ; , , It is an adjustable feedback control coefficient;

[0015] S4, the nonlinear feedback control signal of S3 As a control term, a first-order ordinary differential dynamic system of S2 is introduced to form a controlled dynamic system; the feedback control coefficient is adjusted... , , By changing the critical conditions of the saddle node bifurcation determined by S2 in the controlled dynamic system, the occurrence position of the saddle node bifurcation in the parameter space can be actively adjusted to suppress or delay traffic loss and congestion.

[0016] Compared with the prior art, the present invention has the following advantages:

[0017] 1. Existing macroscopic models, such as LWR and Aw-Rascle, typically rely only on local density or average speed, neglecting the driver's perception of the motion of multiple vehicles ahead. While traditional microscopic models consider multiple vehicles ahead, they struggle to directly derive continuous equations suitable for system-level regulation. This approach incorporates speed difference information from multiple vehicles ahead through a full speed difference model, and through macroscopic continuous derivation, obtains spatiotemporal regulation equations containing higher-order nonlinear terms (such as density gradient and speed difference coupling terms). Compared to macroscopic models that rely solely on information from a single vehicle ahead, this coupled modeling more realistically reflects the impact of actual driving behavior on traffic flow stability, significantly improving the prediction accuracy of critical instability conditions (such as saddle-joint bifurcation points), and laying a physical foundation for subsequent precise intervention.

[0018] 2. Unlike existing research methods that rely on numerical scanning or empirical thresholds to determine the instability boundary, this scheme reduces the partial differential equation to a function of density ρ and its derivative through traveling wave transformation z=x−ct. A first-order ordinary differential system, and at the equilibrium point The Jacobian matrix is ​​calculated at the point of origin, and the critical parameter relationship is explicitly derived using the saddle-node bifurcation criterion of zero determinant. This avoids the critical density deviation problem caused by neglecting higher-order nonlinear terms in traditional methods, making the bifurcation boundary analytically expressible and updatable online. Compared to mainstream control strategies that only focus on Hopf bifurcation, this scheme focuses on more destructive density-abrupt instability, i.e., sudden collapse of free flow, achieving targeted identification of more dangerous instability modes.

[0019] 3. Existing feedback control methods mostly employ linear forms, which are difficult to compensate for the inherent strong nonlinear characteristics of the system; while this scheme innovatively introduces a squared term of the disturbance x. This design constructs a nonlinear feedback structure, causing the control action to nonlinearly increase with the degree of deviation from the equilibrium point. Compared to pure linear feedback, this design can more effectively offset the lack of nonlinear restoring force in the traffic flow model, such as acceleration response saturation when density increases, and provides stronger reverse traction force near the bifurcation point to prevent the system from slipping into unstable branches. Especially when the disturbance amplitude is large, its suppression capability is significantly better than that of a linear controller, avoiding over-adjustment or under-adjustment.

[0020] 4. Control signal As a new input to the controlled system, the bifurcation critical condition of the original system is transformed from a fixed parameter into an adjustable function—through real-time adjustment. , , This method can proactively shift the critical density at which the saddle-shaped bifurcation occurs, thus maintaining a uniform flow stability even under higher traffic loads. Unlike traditional passive management methods such as flow restriction and diversion, this method proactively reshapes the system's inherent instability mechanism. Compared to existing technologies that only address congestion after it occurs, this strategy can intervene before the traffic density reaches historical critical values, fundamentally delaying or even preventing instability transitions, demonstrating significant foresight and preventative capabilities.

[0021] In summary, this method overcomes the limitations of traditional traffic control methods that rely on empirical thresholds and linear assumptions. By deeply integrating nonlinear dynamics theory into real-time intervention decision-making, it significantly improves the ability to prevent sudden and irreversible congestion. This method can achieve source suppression before congestion occurs.

[0022] Preferably, in S2, the critical condition for saddle node bifurcation in the system is: traveling wave parameters ;in, For the equilibrium velocity-density function density The first derivative, For its in The derivative at point .

[0023] With this setup, the critical condition is directly related to the measurable / predictable physical quantity in a closed-loop analytical form, and the equilibrium density... With the slope of the velocity-density curve This avoids the high computational cost and lag of traditional numerical methods that require repeated simulations to scan the parameter space; more importantly, it reveals that the instability threshold is essentially determined by the local curvature of the traffic flow constitutive relation—when More negative, meaning the velocity decreases faster with increasing density, critical. As the absolute value increases, the system becomes more prone to instability; conversely, it becomes more robust. This clear physical explanation provides a clear target basis for the control strategy, significantly improving the design rationality and real-time adaptability of intervention measures, and providing a quantifiable and online-updable instability early warning benchmark for subsequent nonlinear feedback control (S3–S4).

[0024] Preferably, the equilibrium velocity-density function for:

[0025] ;

[0026] in, For free flow velocity, This represents the maximum traffic density.

[0027] With this configuration, the function significantly improves its interpretation of velocity gradients near the critical density while maintaining the physical interpretability of the macroscopic model. Modeling accuracy—especially in uniform flow steady state In the neighborhood, its first derivative The numerical values ​​more closely approximate the nonlinear characteristic of "speed decreasing at an accelerated rate with increasing density" reflected in measured traffic data; compared to the traditional Greenshields linear model or the uncorrected Logistic model, this form can more accurately predict the critical conditions for saddle-node bifurcation. Key factors in This avoids overestimation or underestimation of critical density due to distortion of the speed-density relationship, ensuring that the nonlinear feedback control (S3–S4) designed based on this critical condition has reliable triggering timing and control intensity in real traffic scenarios, providing a solid data foundation for source-based congestion suppression.

[0028] Preferably, in S1, the micro-carriage following model is expressed as:

[0029] ;

[0030] in, Let be the speed of the nth vehicle at time t; V is the sensitivity coefficient; V(·) is the optimization velocity function. For the distance between the front of the car, The response coefficient is given by the speed difference of the vehicle immediately ahead, where m is the number of vehicles the driver is paying attention to ahead. To calculate the speed difference of the kth vehicle ahead The sensitivity coefficient.

[0031] With this setup, 1. the model explicitly introduces a weighted superposition term of the speed differences of multiple preceding vehicles, based on the traditional full speed difference model. And assign a differentiated sensitivity coefficient to each preceding vehicle. Instead of using uniform weights, this approach more realistically portrays the dynamic behavior of drivers allocating their attention to vehicles at different distances in actual driving. For example, the deceleration signals of the second or third vehicle ahead may produce a delayed but cumulative response. This improvement enables the model to reproduce the amplification effect of traffic oscillations caused by coordinated deceleration of multiple vehicles in reality, significantly enhancing its simulation capability for the initial stage of congestion waves—that is, the evolution of small disturbances into macroscopic instability. This provides a high-fidelity microscopic foundation for subsequent macroscopic continuous derivation and avoids deviations in critical instability conditions due to neglecting the influence of distant vehicles ahead.

[0032] 2. By splitting the speed difference response term into the dominant term immediately preceding the vehicle. Coupled with multiple front vehicles This model achieves a clear decoupling of the physical meaning of the parameters: λ can be calibrated as the short-time response intensity, while γk reflects the driver's ability to anticipate and trust in distant vehicles. This not only facilitates parameter identification based on measured trajectory data but also provides an interpretable mapping basis for the design of nonlinear feedback control coefficients in S3, enhancing the adjustability and engineering feasibility of the entire control framework.

[0033] Preferably, in S1, the macroscopic continuity derivation includes:

[0034] 1) Establish the mapping relationship from micro variables to macro variables, including:

[0035] , , , ;

[0036] in, This represents the macroscopic velocity field at spatial location x and time t under the assumption of a macroscopic continuum. represents the speed of the (n+1)th vehicle at time t; Δ represents the average distance between adjacent vehicles. This represents the macroscopic velocity field at a distance Δ in front of position x at time t; For the speed of perturbation propagation; The viscosity coefficient;

[0037] 2) Based on the mean-value theorem, the micro velocity difference... The mapping is approximated by a macroscopic expression, and the relationship is as follows:

[0038] ;

[0039] in, This represents the speed difference between the nth car and the car immediately preceding it at time t; Let x be a point in the interval [x, x+Δ]. For macroscopic velocity field In position The incremental form of the first-order partial derivative with respect to spatial position x; For macroscopic velocity field The first partial derivative with respect to spatial position x; For macroscopic velocity field The second partial derivative with respect to spatial position x;

[0040] Substituting the mapping and approximation relationships from steps 1) and 2) into the micro-car-following model, the macro-traffic flow dynamics equation is derived.

[0041] With this setup, 1. the derivation achieves a rigorous asymptotic match from discrete race-following dynamics to continuous partial differential equations while preserving higher-order spatial nonlinear effects—by introducing The term explicitly incorporates curvature correction due to the discreteness of vehicle spacing, avoiding the systematic bias caused by using only first-order Taylor expansion in traditional macroscopic modeling; especially when traffic density increases and vehicle spacing decreases, i.e. Δ becomes smaller but... As the value increases, the contribution of the second-order term to the velocity gradient becomes significant. Its inclusion enables the derived macroscopic equations to accurately reproduce the shock precursors and instability evolution paths caused by local velocity discontinuities at the microscopic level, providing a mathematically consistent and physically reliable continuous model basis for the subsequent accurate solution of the saddle-knot bifurcation critical condition (S2).

[0042] 2. By mapping discrete parameters λ and Δ to macroscopic physical quantities and This scheme establishes a calibrable bridge between microscopic driving behavior parameters and macroscopic fluid analogy parameters: for example, It can be inverted from the measured propagation speed of the congestion wave. It can correlate driver reaction delay with car-following smoothness; this makes the final macroscopic equation not only theoretically rigorous, but also supports online parameter calibration based on field data, significantly enhancing the model's adaptability and prediction reliability in real road networks, and providing a dynamically updated system state benchmark for real-time intervention strategies (S3–S4).

[0043] Preferably, in S2, the first-order ordinary differential dynamical system is defined by the following set of equations:

[0044] ;

[0045] in, and For density Wave velocity c and traveling wave parameters The function;

[0046] ;

[0047] ;

[0048] in, For the speed of perturbation propagation; Here, is the viscosity coefficient; Δ represents the average distance between adjacent vehicles. It is the equilibrium velocity-density function.

[0049] With this setup, the original high-dimensional nonlinear partial differential problem of the first-order differential system is precisely reduced to a two-dimensional phase plane dynamics problem, making the system's stability and bifurcation behavior possible by analyzing the equilibrium points. The eigenvalues ​​of the Jacobian matrix are used for direct determination; especially crucial are the right-hand side terms. Middle of the content and The coupling structure makes the critical condition The derivation is based on rigorous mathematical evidence. This structure ensures the complete closure of the causal chain from microscopic racing behavior and macroscopic fluid properties to the critical instability threshold, avoiding uncertainties caused by empirical assumptions or numerical fitting. It provides a unique, definite, and analytically verifiable instability criterion for the subsequent design of active intervention strategies based on bifurcation criticality (such as nonlinear feedback in S3).

[0050] Preferably, in S2, at the equilibrium point At this point, the elements of the Jacobian matrix are determined by the following formula:

[0051] ;

[0052] ;

[0053] in, and These are the elements at the corresponding positions in the Jacobian matrix; Let i represent the traffic density of road segment i. For the equilibrium velocity-density function density The first derivative; For its in The derivative at point .

[0054] This setup directly anchors the critical condition for system instability to the first derivative of the equilibrium velocity-density function at the equilibrium density. And clearly reveal: if and only if When the Jacobian matrix exhibits zero eigenvalues, the system undergoes saddle-node bifurcation—equivalent to the classical critical flow rate condition. .because The critical flow rate can be accurately calculated using the velocity-density function optimized in S1, which gives the expression for the critical flow rate. Instead of relying on fitting or simulation, the density can be measured in actual road sections. With the already calibrated The analytical method yields a rigorous, rapid, and embeddable real-time criterion for dynamically identifying the instability threshold of each road segment in multi-segment coordinated control. This ensures that the intervention strategy (S3) is triggered before actual instability occurs, avoiding delayed or false triggering.

[0055] Preferably, the sensitivity coefficient satisfy 0.

[0056] Preferably, in S3, the nonlinear feedback control signal middle, Control parameters configured to adjust the position of the saddle node bifurcation in parameter space.

[0057] With this setup, the nonlinear control structure achieves explicit parameterization of the bifurcation critical point with minimal order.

[0058] Preferably, in S3, the disturbance amount x of the traffic density is . Attached Figure Description

[0059] To make the objectives, technical solutions, and advantages of the invention clearer, the invention will now be described in further detail with reference to the accompanying drawings, wherein:

[0060] Figure 1 This is a flowchart of the method;

[0061] Figure 2 The phase plane in Example 1 A schematic diagram of the phase track structure;

[0062] Figure 3 For different initial densities in Example 2 Density spacetime plot of values;

[0063] Figure 4 This is a schematic diagram of the Hopf(H) bifurcation and limit point (LP) in Example 2;

[0064] Figure 5 This is a phase plane diagram showing the changes in system state before and after the saddle node bifurcation in Example 2;

[0065] Figure 6 This is a density-space-time diagram of the system state changes before and after the saddle node bifurcation in Example 2;

[0066] Figure 7 This is a diagram showing the system changes before and after the saddle node bifurcation in Example 2. Detailed Implementation

[0067] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0068] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of the present invention, not all of them. Therefore, the following detailed description of the embodiments of the present invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely to represent selected embodiments of the invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without inventive effort are within the scope of protection of the present invention.

[0069] Example 1

[0070] like Figure 1 As shown, this invention provides a traffic flow congestion intervention method based on saddle node bifurcation control, comprising the following steps:

[0071] S1. Based on the Full Velocity Difference (FVD) model, this paper introduces the vehicle's perception of the speed differences between multiple vehicles ahead, constructs a micro-car-following model, and derives a macro-continuous model describing traffic density using spatial location x and time t as independent variables. and macro speed Macroscopic traffic flow dynamics equations of spatiotemporal evolution.

[0072] In practical implementation, the micro-carriage-following model is expressed as follows:

[0073] ;

[0074] in, Let be the speed of the nth vehicle at time t; V is the sensitivity coefficient; V(·) is the optimization velocity function. For the distance between the front of the car, The response coefficient is given by the speed difference of the vehicle immediately ahead, where m is the number of vehicles the driver is paying attention to ahead. To calculate the speed difference of the kth vehicle ahead The sensitivity coefficient. satisfy 0.

[0075] The macroscopic continuity derivation includes:

[0076] 1) Establish the mapping relationship from micro variables to macro variables, including:

[0077] , , , ;

[0078] in, This represents the macroscopic velocity field at spatial location x and time t under the assumption of a macroscopic continuum. represents the speed of the (n+1)th vehicle at time t; Δ represents the average distance between adjacent vehicles. This represents the macroscopic velocity field at a distance Δ in front of position x at time t; For the speed of perturbation propagation; The viscosity coefficient;

[0079] 2) Based on the mean-value theorem, the micro velocity difference... The mapping is approximated by a macroscopic expression, and the relationship is as follows:

[0080] ;

[0081] in, This represents the speed difference between the nth car and the car immediately preceding it at time t; Let x be a point in the interval [x, x+Δ]. For macroscopic velocity field In position The incremental form of the first-order partial derivative with respect to spatial position x; For macroscopic velocity field The first partial derivative with respect to spatial position x; For macroscopic velocity field The second partial derivative with respect to spatial position x;

[0082] Substituting the mapping and approximation relationships from steps 1) and 2) into the micro-car-following model, the macro-traffic flow dynamics equation is derived.

[0083] To facilitate a better understanding of the technical content of method S1 by those skilled in the art, the following explanation is provided.

[0084] The equations for the Full Velocity Difference (FVD) model are as follows:

[0085] (1-1)

[0086] However, the FVD model only considers information from the nearest preceding vehicle, neglecting multi-vehicle visibility and cooperative behavior, and cannot describe the driver's advance reaction and anticipated adjustment behavior. This solution introduces a following model as a foundation to improve the model's predictive accuracy and applicability. The model comprehensively considers the dynamic disturbances caused by speed differences between multiple preceding vehicles and extends existing following theories and research findings, thus constructing a more complete and explanatory framework for describing traffic flow. This model not only provides a new perspective for a deeper understanding of the evolutionary characteristics of traffic flow but also offers valuable reference for the subsequent expansion and optimization of traffic flow theory.

[0087] (1-2)

[0088] in, Represents the sensitivity coefficient. It is the response coefficient. The number of vehicles in front that the driver is paying attention to (e.g.) This indicates that you are paying attention to both the car in front and the car in front of that. For the first The sensitivity coefficient for the speed difference between the two vehicles. Typically, That is, the vehicle directly in front has the greatest impact, and the impact decreases as the vehicle further ahead is further away. The coefficient represents the driver's sensitivity to changes in the speed of vehicles at different positions ahead. For example... A larger value indicates that the driver is paying more attention to the vehicle directly in front; while The value may be small, indicating a lower level of attention to the vehicle directly in front, but it still has some impact. The driver pays attention not only to the vehicle directly in front (the (n+1)th vehicle), but also to the speed changes of vehicles further ahead (the (n+2), (n+3), ..., (n+m)th vehicles). This represents the speed difference between the nth car and the kth car ahead (i.e., the (n+kth)th car). It is the optimal speed function for the headway, and the mathematical expression of the optimal speed function is as follows:

[0089] (1-3)

[0090] Meanwhile, the mapping relationship from the micro model to the macro model is as follows:

[0091] (1-4)

[0092] Here Indicates the distance between consecutive vehicles. and These represent macroscopic density and macroscopic velocity, respectively. It is the equilibrium velocity. According to the mean value theorem, we know that:

[0093] (1-5)

[0094] Substituting the above macroeconomic variables into equation (1-2), we can obtain:

[0095] (1-6)

[0096] By combining the conservation equations in fluid mechanics theory with the new dynamic equations, we can obtain:

[0097] (1-7)

[0098] S2. Introduce a traveling wave transformation z = x - ct into the macroscopic traffic flow dynamics equation of S1, where c is the wave velocity, to transform it into a function relating to traffic density. The second-order ordinary differential equation, further expressed as density and its derivative with respect to the traveling wave coordinate z A first-order ordinary differential dynamical system with state variables; determine the equilibrium point of the first-order ordinary differential dynamical system. ,in The traffic density of the system under uniform steady state is calculated, and its Jacobian matrix is ​​calculated at the equilibrium point. Based on the saddle node bifurcation criterion that the determinant of the Jacobian matrix is ​​zero, the critical condition for the system to undergo saddle node bifurcation is determined.

[0099] In practical implementation, the critical condition for saddle node bifurcation in the system is: traveling wave parameters. ;in, For the equilibrium velocity-density function density The first derivative, For its in The derivative at point .

[0100] The equilibrium velocity-density function for:

[0101] ;

[0102] in, For free flow velocity, This represents the maximum traffic density.

[0103] In practical implementation, the first-order ordinary differential dynamical system is defined by the following set of equations:

[0104] ;

[0105] in, and For density Wave velocity c and traveling wave parameters The function;

[0106] ;

[0107] ;

[0108] in, For the speed of perturbation propagation; Here, is the viscosity coefficient; Δ represents the average distance between adjacent vehicles. It is the equilibrium velocity-density function.

[0109] At the equilibrium point At this point, the elements of the Jacobian matrix are determined by the following formula:

[0110] ;

[0111] ;

[0112] in, and These are the elements at the corresponding positions in the Jacobian matrix; Let i represent the traffic density of road segment i. For the equilibrium velocity-density function density The first derivative; For its in The derivative at point .

[0113] To facilitate a better understanding of the technical content of method S2 by those skilled in the art, the following explanation is provided.

[0114] To better reflect actual traffic flow characteristics and improve the model's interpretability, the main roads are assumed to have open boundary conditions.

[0115] (2-1)

[0116] Assume the model has a traveling wave solution. and ,in , Indicates wave speed, and Substituting these solutions into equation (1-7), we can derive the following result:

[0117] (2-2)

[0118] (2-3)

[0119] From equation (2-3), we can obtain The equation Differentiating both sides with respect to z, we get .

[0120] (2-4)

[0121] (2-4)

[0122] Substituting equations (2-4) and (2-5) into equation (2-3), we get:

[0123] (2-6)

[0124] By integrating formula (2-2), we can obtain:

[0125] (2-7)

[0126] Substituting equation (2-7) into equation (2-6), we get:

[0127] (2-8)

[0128] After performing the transformation, we can obtain:

[0129] (2-9)

[0130] After simplification, we can obtain a result about The second-order ordinary differential equation:

[0131] (2-10)

[0132] In this formula, where:

[0133] (2-11)

[0134] (2-12)

[0135] set up Then equation (2–10) can be transformed into a system of first-order ordinary differential equations:

[0136] (2-13)

[0137] By setting the right-hand side of the equation to zero, we get:

[0138] (2-14)

[0139] Therefore, its equilibrium point can be determined. At the equilibrium point, performing a Taylor expansion on equation (2-11) yields the linear system shown in equation (2-13):

[0140] (2-15)

[0141] Therefore, the above equation at the equilibrium point The Jacobian matrix at that point can be represented as:

[0142] (2-16)

[0143] Therefore, the Jacobian characteristic equation is:

[0144] (2-17)

[0145] in , .from and ,get:

[0146] (2-18)

[0147] (2-19)

[0148] Because at the equilibrium point ,therefore ,then It can be rewritten as:

[0149] (2-20)

[0150] According to the theory of differential equations, the type of equilibrium point of the linear system (2-13) can be determined as follows:

[0151] (a) When At this point, the equilibrium point is the saddle point;

[0152] (b) when and At that time, the equilibrium point is a node;

[0153] (c) when and At that time, the equilibrium point is the focus;

[0154] (d) when and At that time, the equilibrium point is the center.

[0155] when When the linear system is at a saddle point, its stability is unstable; when (or When ), the stability of nodes or foci is important. (or It is stable.

[0156] According to the Hartman-Grobman linearization theorem, the nonlinear system (2-11) and the linear system (2-13) share the same equilibrium point. For non-central equilibrium points, the stability of the nonlinear system (2-11) and the linear system (2-13) is consistent at the equilibrium point. Given any set of traveling wave velocities c and traveling wave parameters... The value of can determine the equilibrium point of the linear system (2-13). (i = 1, 2, 3).

[0157] Figure 2 Give phase plane The phase-track structure is used to illustrate the interaction of the system's equilibrium state under different traveling wave velocities and parameters. Figure 2 (a) Corresponding parameter set (traveling wave velocity (c=-1.371), traveling wave parameters) =0.2), the three red dots on the horizontal axis in the figure correspond to three equilibrium points: the left end For stable nodes (free flow), the middle one For the spiral point (congestion flow) and the right end This is a saddle point (unstable). The solid blue line in the diagram shows: when... At that time, several trajectories can be observed near the saddle point. It then converges in a spiral pattern and eventually points to the focal point. ;when At this point, these trajectories gradually diverge after leaving the focal point, eventually extending to infinity. Their stable and unstable manifolds (indicated by solid green lines) divide the phase plane space into different regions of attraction. Specifically, when At that time, several trajectories starting near the focus evolve along the unstable direction and eventually tend towards the node on the left. That is, the system returns to a low-density free-flow state; while when At that time, the trajectory located on the right side of the stable manifold at the node is guided to the unstable point at the right end. This process manifests as an evolution towards a high-density congestion state accompanied by oscillating decay.

[0158] Figure 2 (b) Corresponding to another parameter set (traveling wave velocity (c=-1.38), traveling wave parameters) =0.3), the phase plane structure undergoes significant changes. Saddle point The stable and unstable manifolds intersect, forming a type of enclosed region; the several closed trajectories shown by the solid green line in the figure form a limiting cycle, with the outer and inner trajectories exhibiting different final fates. Trajectories near the saddle point continue to move away from the unstable flow direction. However, as time evolves, part of the trajectory falls into a closed oscillating loop and oscillates periodically within the loop. Subsequently, it may decay and be drawn towards the helical attractor on the left. It depends on the initial position.

[0159] Stability and bifurcation analysis of the model

[0160] Based on the aforementioned macroscopic traffic flow model and the derived first-order ordinary differential dynamic system (2-11), nonlinear dynamic analysis can be performed on it. Using the traveling wave parameter q* as the key bifurcation parameter of the system, the system at the equilibrium point is analyzed. The properties of the Jacobian matrix (Equation (3-1)) can reveal the mechanism of system instability.

[0161] (3-1)

[0162] Existence and control of saddle node bifurcation

[0163] Lemma: Consider the system , , , It is a variable parameter. If... Satisfying the equilibrium point condition ,remember ,make and They are respectively The left and right unit eigenvectors, i.e. and Then, when the following conditions are met, It is a saddle-type branch of the system.

[0164] (4-1)

[0165] (4-2)

[0166] Then for any small Its solution curve is in The approximate expression for the vicinity is:

[0167] (4-3)

[0168] For system (2-11), let... For , the Jacobian matrix at the equilibrium point is shown in equation (4-1), which is a variable parameter. At that time, there exists satisfy ,at this time,

[0169] (4-4)

[0170] variables and Substituting into equations (4-1) and (4-2), we get:

[0171] (4-5)

[0172] (4-6)

[0173] Therefore when At that time, the system (2-11) was There is a saddle-shaped branch.

[0174] S3. Construct a nonlinear feedback control signal Where x is the traffic density relative to the equilibrium point density; The disturbance quantity; y is the state variable in S2. ; , , It is an adjustable feedback control coefficient.

[0175] In practice, The control parameters are configured to adjust the position of the saddle node bifurcation in the parameter space. The disturbance amount x of the traffic density is... .

[0176] To facilitate a better understanding of the technical content of method S3 by those skilled in the art, the following explanation is provided.

[0177] By combining polynomial approximation methods to handle system uncertainties, an equivalent deterministic system is determined for the stochastic system to achieve bifurcation control. In practical applications, external intervention is applied to the system to achieve optimal control. Based on the proposed nonlinear system (2-11), the nonlinear feedback control signal is defined as:

[0178] (4-7)

[0179] in, , , This is the feedback coefficient.

[0180] First, substitute equations (2-11) and (2-12) into the second equation of equation (2-13), and then simplify it to the following form:

[0181] (4-8)

[0182] To facilitate subsequent derivation, use Alternative density variables And rewrite equation (4-8) as follows:

[0183] (4-9)

[0184] in: ;

[0185] ;

[0186] ;

[0187] Then, the improved control system (called a stochastic feedback control system) incorporates feedback control signals. The system (2-13) becomes:

[0188] (4-10).

[0189] S4, the nonlinear feedback control signal of S3 As a control term, a first-order ordinary differential dynamic system of S2 is introduced to form a controlled dynamic system; the feedback control coefficient is adjusted... , , By changing the critical conditions of the saddle node bifurcation determined by S2 in the controlled dynamic system, the occurrence position of the saddle node bifurcation in the parameter space can be actively adjusted to suppress or delay traffic loss and congestion.

[0190] This method overcomes the limitations of traditional traffic control methods that rely on empirical thresholds and linear assumptions. By deeply integrating nonlinear dynamics theory into real-time intervention decision-making, it significantly improves the ability to prevent sudden and irreversible congestion. This method can achieve source suppression before congestion occurs.

[0191] Example 2

[0192] To better illustrate the effectiveness of this method, the following verification is provided.

[0193] Simulation experiment on the impact of multi-vehicle cooperative feedback on traffic flow

[0194] Simple theoretical analysis often fails to fully reveal the complex phenomena in actual traffic flow. Therefore, the study of saddle-bifurcation analysis and control based on nonlinear dynamic systems with multiple preceding vehicles can verify the accuracy of theoretical analysis and provide more specific practical guidance for traffic flow optimization control. Numerical simulation methods are used to deeply analyze the saddle-bifurcation phenomenon in the model, revealing its generation mechanism and laws, and combining it with practical cases for analysis. This allows for the formulation of corresponding control strategies to achieve optimal traffic flow control. Based on actual traffic flow data, the theoretical results of the saddle-bifurcation phenomenon will be further verified by comparing it with actual conditions. By setting appropriate parameters for comparison, the impact of speed differences among multiple preceding vehicles on the overall traffic flow in complex traffic environments is studied. Traffic flow is simulated to show the driving and congestion patterns in the traffic system, while other parameters remain unchanged. Adding small local disturbances to the initial uniform traffic flow can clearly show the stopping phenomenon when the disturbance is amplified. The expression for the initial density is as follows:

[0195] (5-1)

[0196] (5-2)

[0197] in For the initial density, For perturbation density, The length of the road segment and the dynamic proximity boundary conditions are given by the following formula:

[0198] (5-3)

[0199] For ease of simulation, the spatial spacing is taken as equidistant 100m, the time interval is taken as 1s, and the values ​​of other parameters in the model are as follows:

[0200] (5-4)

[0201] like Figure 3 As shown, under the condition of fixed initial density, different parameters The choice of values ​​leads to significantly different density evolution patterns in the transportation system. Among these parameters... It is the sensitivity coefficient to speed differences between multiple vehicles ahead. It represents the intensity or sensitivity of a vehicle's utilization of information from these distant vehicles ahead. When At times, such as Figure 3 As shown in (a), the initial small disturbance did not disappear but instead rapidly amplified over time, forming a huge peak. The road density field exhibited continuous spatiotemporal oscillations and local congestion clusters, and the system was in an unstable state. At that time, compared to Figure 3 (a) The amplitude of the congestion wave has decreased, the rate of unstable growth has slowed, and the system is on the verge of transitioning from instability to stability. Figure 3 As shown in (b). With When the increase, At this point, the initial disturbance rapidly decays over time, eventually resulting in a smooth density surface, indicating that the traffic flow has returned to a stable, uniform state. Therefore, when... When the volume is small, the system is unstable, and minor disturbances can escalate into severe traffic congestion. When the speed difference between multiple vehicles ahead is increased to a certain threshold, the system can effectively suppress disturbances, eliminate congestion, and keep traffic flow stable and efficient by utilizing the speed difference information between multiple vehicles ahead. As a sensitivity coefficient for multiple preceding vehicle speed differences, it affects the ability to suppress traffic waves and the stability of the system.

[0202] The system can reach multiple equilibrium points by selecting different parameter values. By treating various factors as continuous variable parameters, various system bifurcations can be generated using the bifurcation software package Matcont.

[0203] When parameter When the initial value is set to 0.2, the actual calculation range is... Approximately -30 to 30, within this calculation range, two special points are found: the Hopf(H) bifurcation and the limit point (LP), such as... Figure 4 As shown.

[0204] exist When the value is 0.172588, the saddle node bifurcation point appears, and the limit point state variable... That is, the vehicle density at this time. The two eigenvalues ​​calculated are respectively and The latter eigenvalue is close to 0, which is an important basis for judging that it is a limit point branch. The real part of the other eigenvalue is positive, indicating that the limit point is unstable, that is, an unstable saddle-node bifurcation point.

[0205] exist When the value is 0.762689, the state variable at the limit point is (0.05335, 0), which represents the vehicle density at this point. The two eigenvalues ​​calculated are respectively Here, the real part of a pair of conjugate eigenvalues ​​is considered to be 0, which is a sign that it is a Hopf branch. If one of the eigenvalues ​​has a negative real part, it indicates a stable saddle node bifurcation.

[0206] Figure 5 As shown, the analysis of the above bifurcation points demonstrates the stability changes of the system when it passes through the saddle node bifurcation point. By analyzing the first LP bifurcation point, i.e., when the traveling wave parameters... Impact on traffic flow when bifurcating at a saddle node. When considering In the case where the system undergoes a bifurcation at a saddle node, let's assume... At times, such as Figure 5 As shown in (a), the system has two equilibrium points: a saddle point and a spiral point. The system is unstable at the equilibrium point (f2,0), but exhibits equilibrium at (f1,0). The trajectory near the equilibrium point is attracted by this point, indicating that the system is in a stable state at (f1,0). At times, such as Figure 5 As shown in (b), the two equilibrium points gradually move towards the center. At that time, and Figure 5 Compared to the equilibrium point in (b), Figure 5 In (c), one equilibrium point disappears, and a saddle point appears after the two equilibrium points collide. At this point, the system experiences saddle-node bifurcation. Near the saddle point, the curves converge to the lower right, while away from the saddle point, the system is in an unstable state, such as... Figure 5 As shown in (c). When At that time, with The increase in the value of the flow means that after the system bifurcates at the saddle node, all equilibrium points disappear, and the flow becomes unstable, such as... Figure 5 As shown in (d).

[0207] In addition, density spatiotemporal diagrams can be used to monitor the state evolution of the system during the bifurcation process at the saddle node. By comparing the graphical features on both sides of the bifurcation point, the significant changes in the internal stability structure of the system after the parameters reach a specific threshold can be intuitively identified.

[0208] like Figure 6 As shown, taking the second LP bifurcation point as an example, the system parameters at this time are... The saddle node coordinates are (0.05335, 0), and the vehicle density is... The stability changes of the system when it undergoes bifurcation at a saddle node are analyzed. The initial density before bifurcation... At that time, the initial density disturbance was not attenuated by the system, but instead continued to expand during the spatiotemporal evolution, forming a continuously propagating stop-and-go traffic wave and a multi-peak congestion cluster. The density spatiotemporal surface exhibited significant spatiotemporal oscillations and multiple longitudinal density peaks and ridges, indicating that the system was in an unstable state; while when the initial density increased to When the system passes the saddle node bifurcation point, the initial disturbance is rapidly absorbed and attenuated by the system, the density surface quickly flattens, and the overall density returns to a uniform equilibrium state. This indicates that the system's stability structure has undergone a fundamental change and it is in a stable state. It can be concluded that the relative magnitude of the initial density and the critical density at the saddle node bifurcation point affects the stability of the traffic flow system. When the initial density is below this critical threshold, the system maintains stable operation; when the initial density exceeds the corresponding saddle node bifurcation density, the system will enter an unstable state due to the change in the equilibrium structure, exhibiting unstable traffic flow characteristics.

[0209] Simulation experiment of saddle joint control with multiple front vehicle speed difference

[0210] When a system experiences a saddle-node bifurcation, it often exhibits nonlinear characteristics such as abrupt amplitude changes and response delays during phase transitions. This phenomenon can disrupt the system's original equilibrium state. To prevent such destructive dynamic behaviors, researchers typically intervene or regulate the system in various ways to maintain traffic flow stability and suppress congestion. Adjusting the nonlinear feedback controller... Medium parameters The value of can effectively change the location of the saddle-knot bifurcation point, thereby controlling unstable traffic flow, that is, adjusting the timing of the saddle-knot bifurcation phenomenon. To further verify the regulatory effect of the feedback controller, the control coefficient is used... As the research subject, while maintaining Under the condition that remains unchanged, the bifurcation diagram of the controlled system was drawn, as follows: Figure 7 As shown, the results indicate that with The position of the saddle-node bifurcation point also shifts with the change in the parameters, thus verifying the significant impact of feedback control parameters on system stability.

[0211] Depend on Figure 7 The results show that when the control parameter When the parameters are 15, 10, 6, and 4.5, the saddle-node bifurcation point of the system shows a gradual rightward shift, indicating that the bifurcation position shifts significantly with parameter changes. In other words, as the parameters change... The adjustment resulted in the saddle-node bifurcation point shifting forward or backward in the phase diagram. This result fully demonstrates that the designed feedback controller can effectively adjust the position of the saddle-node bifurcation point, thereby achieving active control of the system's bifurcation behavior and verifying its feasibility and effectiveness in suppressing traffic flow instability.

[0212] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit the technical solutions. Those skilled in the art should understand that any modifications or equivalent substitutions to the technical solutions of the present invention without departing from the spirit and scope of the present invention should be covered within the scope of the claims of the present invention.

Claims

1. A traffic flow congestion intervention method based on saddle node bifurcation control, characterized in that, Includes the following steps: S1. Based on the full speed difference model, the perception of speed differences between multiple vehicles ahead is introduced to construct a micro-car-following model. Through macroscopic continuous derivation of the micro-car-following model, a model describing traffic density is obtained, with spatial location x and time t as independent variables. and macro speed Macroscopic traffic flow dynamics equations of spatiotemporal evolution; S2. Introduce a traveling wave transformation z = x - ct into the macroscopic traffic flow dynamics equation of S1, where c is the wave velocity, to transform it into a function relating to traffic density. The second-order ordinary differential equation, further expressed as density and its derivative with respect to the traveling wave coordinate z A first-order ordinary differential dynamical system with state variables; determine the equilibrium point of the first-order ordinary differential dynamical system. ,in The traffic density of the system under uniform steady state is calculated, and its Jacobian matrix is ​​calculated at the equilibrium point. Based on the saddle node bifurcation criterion that the determinant of the Jacobian matrix is ​​zero, the critical condition for the system to undergo saddle node bifurcation is determined. S3. Construct a nonlinear feedback control signal Where x is the traffic density relative to the equilibrium point density; The disturbance quantity; y is the state variable in S2. ; , , It is an adjustable feedback control coefficient; S4, the nonlinear feedback control signal of S3 As a control term, a first-order ordinary differential dynamic system of S2 is introduced to form a controlled dynamic system; the feedback control coefficient is adjusted... , , By changing the critical conditions of the saddle node bifurcation determined by S2 in the controlled dynamic system, the occurrence position of the saddle node bifurcation in the parameter space can be actively adjusted to suppress or delay traffic loss and congestion.

2. The traffic flow congestion intervention method based on saddle node bifurcation control as described in claim 1, characterized in that, In S2, the critical condition for saddle node bifurcation in the system is: traveling wave parameters. ;in, For the equilibrium velocity-density function density The first derivative, For its in The derivative at point .

3. The traffic flow congestion intervention method based on saddle node bifurcation control as described in claim 2, characterized in that, The equilibrium velocity-density function for: ； in, For free flow velocity, This represents the maximum traffic density.

4. The traffic flow congestion intervention method based on saddle node bifurcation control as described in claim 1, characterized in that, In S1, the micro-carriage following model is expressed as: ； in, Let be the speed of the nth vehicle at time t; V is the sensitivity coefficient; V(·) is the optimization velocity function. For the distance between the front of the car, The response coefficient is given by the speed difference of the vehicle immediately ahead, where m is the number of vehicles the driver is paying attention to ahead. To calculate the speed difference of the kth vehicle ahead The sensitivity coefficient.

5. The traffic flow congestion intervention method based on saddle node bifurcation control as described in claim 4, characterized in that, In S1, the macroscopic continuity derivation includes: 1) Establish the mapping relationship from micro variables to macro variables, including: ， ， ， ； in, This represents the macroscopic velocity field at spatial location x and time t under the assumption of a macroscopic continuum. represents the speed of the (n+1)th vehicle at time t; Δ represents the average distance between adjacent vehicles. This represents the macroscopic velocity field at a distance Δ in front of position x at time t; For the speed of perturbation propagation; The viscosity coefficient; 2) Based on the mean-value theorem, the micro velocity difference... The mapping is approximated by a macroscopic expression, and the relationship is as follows: ； in, This represents the speed difference between the nth car and the car immediately preceding it at time t; Let x be a point in the interval [x, x+Δ]. For macroscopic velocity field In position The incremental form of the first-order partial derivative with respect to spatial position x; For macroscopic velocity field The first partial derivative with respect to spatial position x; For macroscopic velocity field The second partial derivative with respect to spatial position x; Substituting the mapping and approximation relationships from steps 1) and 2) into the micro-car-following model, the macro-traffic flow dynamics equation is derived.

6. The traffic flow congestion intervention method based on saddle node bifurcation control as described in claim 4, characterized in that, In S2, the first-order ordinary differential dynamical system is defined by the following set of equations: ； in, and For density Wave velocity c and traveling wave parameters The function; ； ； in, For the speed of perturbation propagation; Here, is the viscosity coefficient; Δ represents the average distance between adjacent vehicles. It is the equilibrium velocity-density function.

7. The traffic flow congestion intervention method based on saddle node bifurcation control as described in claim 6, characterized in that, In S2, at the equilibrium point At this point, the elements of the Jacobian matrix are determined by the following formula: ； ； in, and These are the elements at the corresponding positions in the Jacobian matrix; Let represent the traffic density of road segment i; For the equilibrium velocity-density function density The first derivative; For its in The derivative at point .

8. The traffic flow congestion intervention method based on saddle node bifurcation control as described in claim 4, characterized in that, The sensitivity coefficient satisfy 0.

9. The traffic flow congestion intervention method based on saddle node bifurcation control as described in claim 1, characterized in that, In S3, the nonlinear feedback control signal middle, Control parameters configured to adjust the position of the saddle node bifurcation in parameter space.

10. The traffic flow congestion intervention method based on saddle node bifurcation control as described in claim 1, characterized in that, In S3, the disturbance x of the traffic density is .

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