Linear Stability Analysis Method Research Articles

Instability of bubbly jets subjected to unrelaxed axial elastic tension

Injection of bubbly jets is a critical process in various fields, such as fuel atomization and petrochemicals. However, existing instability models for bubbly jets have predominantly focused on density variations due to bubble compressibility, largely neglecting the impact of bubbles on the macroscopic rheological properties of the mixture. In this study, we have integrated compressibility with non-Newtonian properties induced by bubbles to explore the evaluation of jet perturbations. Additionally, considering bubble pre-deformation before injection, we particularly examine the role of unrelaxed axial elastic tension in the bubbly jets. Both linear stability analysis and energy budget methods are used to illustrate the contributions of viscosity, elasticity, and compressibility. The results indicate that the viscoelastic effects induced by bubbles can significantly outweigh the effects of compressibility during the jet destabilization process. Notably, a slight unrelaxed tension can markedly increase both the perturbation growth rate and the cutoff wavenumber, potentially making it a more dominant driver of jet instability than the classical surface tension.

Linear Time-Periodic theory-based novel stability analysis method for voltage-source converter under unbalanced grid conditions

Small-signal stability analysis is essential for the safe operation and parameter design of power electronic-dominated grids. Unbalanced conditions introduce more frequency-coupling terms, which are unneglectable in the stability assessment. This paper proposes a novel Linear Time-Periodic (LTP) theory-based stability analysis method and focuses on grid-connected voltage-source converters (VSCs) without improved unbalancing controls. Firstly, the analysis of fundamental solutions of the LTP system reveals that each LTP mode is characterized by a unique damping factor and multiple oscillation frequencies, defined by LTP eigenvalues and related transformation vectors, i.e., (λi,Vi(t)). Then, a method for calculating (λi,Vi(t)) is proposed using the characteristic matrix of the Harmonic State-Space (HSS) model. An iterative sorting method based on the time-domain interpretation of HSS eigenvalues/eigenvectors is proposed to determine accurate (λi,Vi(t)) in the case of the minimum truncation order. Finally, generalized definitions and calculation methods of the widely used indicators for modal analysis are presented to assess system stability and guide the parameter design. Numerical and simulation results verify the proposed stability analysis method and indicate its advantages compared to the existing Floquet and HSS methods.

Gap solitons of spin–orbit-coupled Bose–Einstein condensates in a Jacobian elliptic sine potential

Gap solitons of quasi-one-dimensional Bose–Einstein condensate loaded in a Jacobian elliptic sine potential with spin–orbit-coupled are investigated theoretically. Under the mean-field approximation, the dynamical behaviors of such a system are described using the Gross–Pitaevskii equation (GPE). Firstly, we linearize the GPE to obtain the band-gap structures. Secondly, the Newton-Conjugate-Gradient (NCG) method is used to search for gap solitons. Thirdly, the dynamic stability of the gap solitons obtained by the NCG method was investigated using linear stability analysis and nonlinear dynamic evolution methods. It is found that in this nonlinear system, there exist both stable gap solitons and structurally rich unstable gap solitons. The external potential modulus and the strength of the spin–orbit coupling are important parameters that influence the stability of the gap solitons.

Establishment of Linear Stability Analysis Method of Hypersonic Boundary Layer Reflecting Porous Surface Effect

Establishment of Linear Stability Analysis Method of Hypersonic Boundary Layer Reflecting Porous Surface Effect

A new very high-order finite-difference method for linear stability analysis and bi-orthogonal decomposition of hypersonic boundary layer flow

Precisely predicting laminar-turbulence transition locations is essential for improvements in hypersonic vehicle design related to flow control and heat protection. Currently state-of-the-art eN prediction method requires the evaluation of discrete normal modes F and S for the growth rate of instability wave. Meanwhile, in receptivity studies, both the discrete and continuous modes, including acoustic, entropy, and vorticity modes, contribute to the generation of the initial disturbance. The purpose of this paper is to introduce a new very high-order numerical method to accurately compute these normal modes with finite-difference on a non-uniform grid. Currently, numerical methods to obtain these normal modes include two major approaches, the boundary value problem approach and the initial value problem approach. The boundary value approach used by Malik (1990) [17] deploys fourth-order finite difference and spectral collocation methods to solve a boundary value problem for linear stability theory (LST). Nonetheless, Malik's presentation only demonstrated the computation of discrete modes, but not the continuous modes essential for conducting modal analysis on receptivity data. To obtain the continuous spectrum for his multimode decomposition framework, Tumin (2007) [16] relies on an initial value approach based on the Runge Kutta scheme with the Gram-Schmidt orthonormalization. However, the initial value approach is a local method that does not give a global evaluation of the eigenvalue spectra of discrete modes. Furthermore, Gram-Schmidt orthonormalization, which can be error-prone in implementation, is required at every step of the integration to minimize the accumulation of numerical errors. To overcome the drawbacks of these two approaches, this paper improves the boundary value approach by introducing a new general very high-order finite difference method for both discrete and continuous modes eigenfunctions. This general high-order finite difference method is based on a non-uniform grid method proposed by Zhong and Tatineni (2003) [22]. Under the finite difference framework, discrete and continuous modes can be obtained by imposing proper freestream asymptotic boundary conditions based on the freestream fundamental solution behavior. This asymptotic boundary condition is used for obtaining both discrete and continuous modes that have both distinct (acoustic) and similar (vorticity and entropy) eigenvalues. Extensive verification of the new method has been carried out by comparing the computed discrete and continuous modes. Subsequently, the discrete and continuous modes obtained with this finite difference method are essential for the bi-orthogonal decomposition, which holds promising potential in obtaining an accurate evaluation of receptivity coefficients. The result of the bi-orthogonal decomposition for a hypersonic boundary layer flow over a flat plate is verified by comparing with existing results. Ultimately, the bi-orthogonal decomposition using the eigenfunctions has been applied to a case of freestream receptivity simulation for an axis-symmetric hypersonic flow over a blunt nose cone with modal contributions computed as coefficients for receptivity analysis.

Complex pattern evolution of a two-dimensional space diffusion model of malware spread

In order to investigate the propagation mechanism of malware in cyber-physical systems (CPSs), the cross-diffusion in two-dimensional space is attempted to be introduced into a class of susceptible-infected (SI) malware propagation model depicted by partial differential equations (PDEs). Most of the traditional reaction-diffusion models of malware propagation only take into account the self-diffusion in one-dimensional space, but take less consideration of the cross-diffusion in two-dimensional space. This paper investigates the spatial diffusion behaviour of malware nodes spreading through physical devices. The formations of Turing patterns after homogeneous stationary instability triggered by Turing bifurcation are investigated by linear stability analysis and multiscale analysis methods. The conditions under the occurence of Hopf bifurcation and Turing bifurcation in the malware model are obtained. The amplitude equations are derived in the vicinity of the bifurcation point to explore the conditions for the formation of Turing patterns in two-dimensional space. And the corresponding patterns are obtained by varying the control parameters. It is shown that malicious virus nodes spread in different forms including hexagons, stripes and a mixture of the two. This paper will extend a new direction for the study of system security theory.

Nonlinear Analysis of an Extended Heterogeneous Lattice Hydrodynamic Model Considering on/off-Ramps

To study the impact of on/off-ramps on traffic stability in heterogeneous traffic flow, a novel lattice hydrodynamic model was presented. The new model’s stability condition was determined using the linear stability analysis method. The theoretical results reveal that traffic flow stability is influenced by the proportion of vehicles with different maximum speeds and safe headway, as well as the presence of on-ramps and off-ramps to a certain degree. Through the approximate perturbation method, the mKdV equation and the kink–antikink solution of the traffic density at the jam area are obtained. In order to verify the effectiveness and feasibility of the new model, numerical simulations were conducted to demonstrate on/off ramp effect and different proportions of vehicles which possess bigger maximum velocity or safe space headway affect the traffic stability. The numerical results indicate that in heterogeneous traffic flow scenarios, increasing the ratio of vehicles which possess bigger maximum velocity or bigger safe space headway can lead to a deterioration of traffic stability. The effect of on-ramp could cause traffic instability, while the effect of off-ramp is beneficial for easing traffic congestion.

Study on the steady-oscillatory transition of three-dimensional (3D) natural convection via Hopf bifurcation

The transition from steady-state flow to periodic oscillatory flow for the natural convection by Hopf bifurcation is investigated in a three-dimensional (3D) cavity. The spectral collocation method (SCM) in combination with the artificial compressibility method (ACM), which is developed by ourselves as a numerical method SCM-ACM with high accuracy, is employed to solve the governing equations directly instead of linear stability analysis method that is commonly used for the research on flow instability. The results show that the amplitude decays exponentially with time and the decay rate is linear with the Grashof number (Gr). The critical Grashof number for steady-oscillatory transition is obtained as Grcr = 3.423 × 106. The dimensionless angular frequency ωcr = 0.24 is also determined by Fourier analysis. In this work, we also examine the heat-momentum interactions within the boundary layers, visualize the periodic oscillations of temperature and velocity amplitudes, and analyze the origin of instability from multiple angles. The results show that large oscillations of velocity and temperature are observed near the isothermal walls. The oscillation is enhanced by the increase of thermal boundary layer thickness and flow velocity at both ends of isothermal walls. The maximum velocity and temperature amplitudes appear at the lower left and upper right corners of the mid-plane (Z = 0.5), where are the origin of instability, and the spanwise walls are almost independent of oscillations. The oscillatory flow of natural convection in three-dimensional cavity originates from the continuously increasing buoyancy force, and its transition occurs by Hopf bifurcation. Moreover, the temperature amplitude exhibits a wavy distribution on the mid-plane (X = 0.5) and strongly depends on the depth Z. These results provide benchmark data for future numerical studies and engineering application.

The seizure classification of focal epilepsy based on the network motif analysis

Due to the complexity of focal epilepsy and its risk for transiting to the generalized epilepsy, the development of reliable classification methods to accurately predict and classify focal and generalized seizures is critical for the clinical management of patients with epilepsy. In order to holistically understand the seizure propagation behavior of focal epilepsy, we propose a three-node motif reduced network by respectively simplifying the focal region, surrounding healthy region and their critical regions as the single node. Because three-node motif can richly characterize information evolutions, the motif analysis method could comprehensively investigate the seizure behavior of focal epilepsy. Firstly, we define a new seizure propagation marker value to capture the seizure onsets and intensity. Based on the three-node motif analysis, it is shown that the focal seizure and spreading can be categorized as inhibitory seizure, focal seizure, focal-critical seizure and generalized seizures, respectively. The four types of seizures correspond to specific modal types respectively, reflecting the strong correlation between seizure behavior and information flow evolution. In addition, it is found that the intensity difference of outflow and inflow information from the critical node (connection heterogeneity) and the excitability of the critical node significantly affected the distribution and transition of the four seizure types. In particular, the method of local linear stability analysis also verifies the effectiveness of four types of seizures classification. In sum, this paper computationally confirms the complex dynamic behavior of focal seizures, and the study of criticality is helpful to propose novel seizure control strategies.

Numerical investigation of the natural transition in boundary layers on underwater axisymmetric bodies with superhydrophobic surfaces

The natural transition in boundary layers on axisymmetric bodies with superhydrophobic surfaces is studied using numerical methods. By boundary layer, we mean the boundary layer around the forebody and around the parallel body of an underwater vehicle. A method of calculating basic laminar flows on axisymmetric bodies with superhydrophobic curved surfaces is proposed, and a method of linear stability analysis for the boundary layers that considers the slip-velocity boundary condition for small disturbances is established. The eN method is used to predict the transition locations of the boundary layers. On a superhydrophobic surface, the slip velocity on the wall increases and then decreases along the streamwise direction, resulting in a maximum value at the forebody. The boundary layer thickness on a superhydrophobic surface becomes smaller than that on an ordinary surface. The superhydrophobic surface stabilizes the boundary layer and thus delays the natural transition. As the slip length on the superhydrophobic surface increases, the unstable zone shrinks, and the transition location moves further downstream, indicating that the delay effect of the superhydrophobic surfaces becomes stronger. As the oncoming flow velocity increases, the transition location on the superhydrophobic surface moves upstream and then downstream, leading to a “dangerous” velocity, at which the transition location is closest to the leading edge. The underlying mechanism of the dangerous velocity phenomenon is discussed: An increasing velocity has both stabilizing effect and destabilizing effect on the boundary layer.

A normal-mode approach for high-speed rarefied plane Couette flow

Based on gas kinetic theory, a linear stability analysis method for low-speed rarefied flows was developed by Zou et al. [“A new linear stability analysis approach for microchannel flow based on the Boltzmann Bhatnagar–Gross–Krook equation,” Phys. Fluids 34, 124114 (2022) and “A novel linear stability analysis method for plane Couette flow considering rarefaction effects,” J. Fluid Mech. 963, A33 (2023)]. In the present study, we extended the method to high-speed rarefied flows using the Bhatnagar–Gross–Krook model. The Chebyshev spectral method is employed to discretize physical space, and the Gauss–Hermite and fourth-order Newton–Cotes quadrature methods are used to discretize velocity space. The fourth-order Newton–Cotes quadrature method was found to have sufficient accuracy for the stability analysis, laying the foundation for future research on hypersonic flows. The stability analysis of compressible rarefied Couette flow showed that acoustic modes are reflected between the wall and the relative sonic line, and the variation in their phase speed and growth rate with the wavenumber is not affected by the Mach number (Ma) and the Knudsen number (Kn). Increasing Kn has a stabilizing effect on both the acoustic and viscous modes, but as Ma increases, the attenuation rate of each mode's growth rate gradually decreases. In subsonic and sonic flows, the least stable viscous mode dominates in the case of small numbers. As Kn increases, the viscous mode gradually dominates over all wavenumber ranges considered in subsonic flow. In sonic flow, mode 1 is dominant in the region beyond the range of small wavenumbers. In supersonic flow, mode 2 is the least stable in the large wavenumber ranges, while mode 1 is the least stable in other wavenumber ranges. At a fixed wavenumber, as Kn increases, the decay rate of the growth rate of mode 2 is the highest. Additionally, under different Knudsen numbers, the growth rates of mode 1, mode 2, and the least stable viscous mode monotonically increase with an increase in Ma, with mode 2 showing the most significant increase.

Solitons in a coupled system of fractional nonlinear Schrödinger equations

Interest in physical systems with fractional derivatives has exploded in the 21st century. Similarly, interest in the localized excitations of nonlinear dynamical systems continues to grow significantly in recent times. In this work, we demonstrate the existence of localized solutions in the nonlinear system of three coupled Schrödinger equations with fractional dispersion, linear coupling, and cubic nonlinearities. We then check the stability of these localized solutions, i.e., solitons, by utilizing the linear stability analysis method and by propagating the perturbed solutions using the split-step fast-Fourier beam propagation method. We also prove that the solitary solutions can be stabilized in this system, thanks to the confining influence of the cross-phase modulation effect. The system is set with the same cross-phase modulation parameter but different self-phase modulation coefficients. We further find that the Lévy index (LI), the propagation constant, and the cross-phase modulation effect markedly affect the profiles and the stability domains of the solitons in the three-coupled system. Furthermore, the amplitude of the component with the strongest self-phase modulation is the largest among the three components, and the variation of the LI, the propagation constant, or the cross-phase modulation coefficient does not affect much this result. Besides the common perturbed propagation, the propagation with modulated LI is also investigated in this work, displaying increased instability when the LI is modulated suddenly, as opposed to stable propagation when the change in the LI is gradual.

Stationary and moving bright solitons in Bose–Einstein condensates with spin–orbit coupling in a Zeeman field

We explore the existence and stability domains of stationary and moving bright solitons in spin–orbit-coupled spin-1 Bose–Einstein condensates in an external Zeeman field. Two families of bell-shape bright solitons (plane-wave (PW) phases) and two families of stripe-shape bright solitons (standing-wave (SW) phases) are obtained. We find a relation between the existence of these bright solitons and the single-particle energy spectrum, as well as the requirements of their existence for atomic interactions. The stability of four families of bright solitons is systematically analyzed using the linear stability analysis method. For two families of PW bright solitons, they are unstable only when the strength of the Zeeman field is above a critical value. The critical value is affected by the spin–orbit coupling and hardly affected by atomic interactions. When the velocities of solitons are zero (nonzero), the critical values of these two families are equal (unequal). For two families of SW bright solitons, their stability domains are identical in the absence of the Zeeman field, and they are unstable when the ferromagnetic interaction is strong. In the presence of the Zeeman field, their stability domains are complementary and complex, and one family of SW bright solitons can exist stably in the area with stronger ferromagnetic interaction. Furthermore, we find the collisions of stable bright solitons with different velocities can generate intriguing dynamics.

A novel linear stability analysis method for plane Couette flow considering rarefaction effects

Following the stability analysis method in classic fluid dynamics, a linear stability equation (LSE) suitable for rarefied flows is derived based on the Bhatnagar–Gross–Krook (BGK) equation. The global method and singular value decomposition method are used for modal and non-modal analysis, respectively. This approach is validated by results obtained from Navier–Stokes (NS) equations. The modal analysis shows that LSEs based on NS equations (NS-LSEs) begin to fail when the Knudsen number ($Kn$) increases past$\sim$0.01, regardless of whether a slip model is used. When$Kn\geq 0.01$, the growth rate of the least stable mode is generally underestimated by the NS-LSEs. Under a fixed wavenumber, the pattern (travelling or standing wave) of the least stable mode changes with$Kn$; when the mode presents the same pattern, the growth rate decreases almost linearly with increasing$Kn$; otherwise, rarefaction effects may not stabilize the flow. The characteristic lengths of the different modes are different, and the single-scale classic stability analysis method cannot predict multiple modes accurately, even when combined with a slip model and even for continuum flow. However, non-modal analysis shows that this error does not affect the transient growth because modes with small growth rates offer little contribution to the transient growth. In rarefied flow, as long as the Mach number ($Ma$) is large enough, transient growth will occur in some wavenumber ranges. The rarefaction effect plays a stabilizing role in transient growth. The NS-LSEs-based method always overestimates the maximum transient growth.

Modulational instability in lossless left-handed metamaterials in nonlinear Schrödinger equation with non-integer dimensional space

Apply style for article title, author, affiliation and email as per stylesheet. Several decades ago, antennas had simple shapes that were described in Euclidean geometry. Nowadays, scientists try to make the structure of fractal geometry for applications in the field of electromagnetism, which has led to the development of new innovative antenna devices. Non-integer dimensional space (NDS) is useful to describe the concept of fractional space in fractal structure for real phenomenon of electromagnetic wave propagation. In this work, we investigate effects of NDS and normalized frequency on modulational instability (MI) gain in lossless left-handed metamaterials (LHM). We derive the nonlinear Schrödindiger equation (NLSE) with non-integer transverse laplacian. By means of linear stability analysis method, MI gain expression is also determined. Different forms of figures are obtained due to the signs of group velocity dispersion (GVD) and defocusing/focusing nonlinearity. We show how the increasing value of the normalized frequency enhances the amplitude as well as the bandwidth of MI gain, and waves are more unstable due to non-integer dimension. The obtained results are new and have a relatively newer application in telecommunication by constructing the fractal-shaped antennas operating in multi-frequency bands.

Abundant explicit and exact solutions for the space–time fractional Vakhnenko–Parkes model in the relaxing medium with stability analysis

In this work, we have considered the beta-fractional derivative form of the Vakhnenko–Parkes equation (VPE) to study its novel analytical solutions. To achieve the required new distinct traveling wave solutions, the new sub-equation method is used for the space–time fractional VPE. The dynamics of these solutions are analyzed under different parametric and fractional conditions by the graphical view. The produced wave patterns include combined bright–dark, w-shape, u-shape, v-shape, bright, propagation of two solitons, combined dark–bright, bell shape, a dark, periodic wave, breather, v-shape with two bright spots and combined anti-peakon–bright solutions. In recognizing the physical significance of the obtained wave solutions at each fraction of an interval, these solutions are highly commendable. In addition, to show the stability of the considered model, the stability analysis of the governing model is discussed by the linear stability analysis method which states the behavior of the model in the corresponding medium. These solutions are useful to study the proliferation of high-frequency waves in the relaxing medium.

Two-dimensional spatial optical solitons in Rydberg cold atomic system under the action of optical lattice

Realizing stable high-dimensional light solitons is a long-standing goal in the study of nonlinear optical physics. However, in high-dimensional space, the light field will inevitably be distorted due to diffraction. In order to solve the diffraction effect in nonlinear Kerr media and achieve the spatial localization of light fields, we propose a scheme to generate stable two-dimensional (2D) solitons in a cold Rydberg atomic system with a Bessel optical lattice, where a three-level atomic structure, a weak probe laser field, and a strong control field constitute the Rydberg-dressed atomic system. When the local nonlinearity, Bessel potential, and nonlocal nonlinearity which is caused by the long-range Rydberg-Rydberg interaction (RRI) between Rydberg atoms are balanced, the probe field can be localized. Under the approximation of electric dipole and rotating wave, the stable solution of probe field is obtained by solving Maxwell-Bloch equations numerically. A cluster of 2D spatial solitons, including fundamental, two-pole, quadrupole and vortex solitons, is found in this system. Among them, the fundamental, dipole and quadrupole have, one, two, and four intensity centers, respectively. Vortex solitons, on the other hand, exhibit vertical characters in profiles and phase structures. The formation and transmission of these solitons can be controlled by system parameters, such as the propagation coefficient, the degree of nonlocal nonlinearity, and Bessel lattice strength. The stable regions of these solitons are determined by anti Vakhitov Kolokolov (anti-VK) criterion and linear stability analysis method. It is found that four kinds of solitons can be generated and stably propagate in space with proper parameters. Owing to the different structures of the poles, the fundamental state and vortex state remain stable, while the quadrupole ones are unstable. In the modulation of solitons, there is a cutoff value of propagation constant <inline-formula><tex-math id="M1">\begin{document}${b_{{\text{co}}}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10-20230096_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10-20230096_M1.png"/></alternatives></inline-formula>, only below which value, the solitons can propagate stably. The light intensity of soliton shows a periodic behavior by tuning Bessel lattice strength. The period of the intensity decreases with the order of the solitons as a result of the interaction between the poles. It is also found that the solitons are more stable with weak nonlocal nonlinearity coefficient. This study provides a new idea for the generation and regulation of optical solitons in high dimensional space.

Surface gap solitons and their stabilities in quasi-1D Bose-Einstein condensate with three-body interactions

The dynamical stability properties of surface gap solitons in quasi-one-dimensional Bose-Einstein condensate loaded in the interface between uniform media and a semi-infifinite Jacobian elliptic sine potential with three-body interactions are investigated numerically. Under the mean-fifield approximation, the dynamical behaviors can be well-described by the nonlinear cubic-quintic Gross-Pitaevskii equation. Firstly, many kinds of surface gap solitons, including the surface bright solitons, surface kink solitons and surface bubble solitons, are obtained numerically by the Newton-conjugate gradient method. The surface bright solitons can be excited in the gap only for the case that the chemical potential is negative and their power is beyond a threshold value. All of them are not bifurcated from the Bloch band. A class of surface solitons with new structures, named the surface dark solitons, can be formed when the three-body interactions are taken into account. The surface dark solitons can exist not only in gap but also in band. The numerical results indicate that the amplitude of the surface gap solitons decreases as the three-body interaction strength increases. Both linear stability analysis and nonlinear dynamical evolution methods are applied to investigate the stability properties of surface gap solitons. For surface bright solitons in the semi-infinite gap, there is a critical value when the chemical potential is given. The surface bright solitons become linearly stable as the three-body interaction exceeds the critical value, or they are linearly unstable. Therefore, the three-body interaction strength plays an important role on the stability of surface gap solitons. One can change the dynamical behaviors of surface gap solitons by adjusting the three-body interaction strength in experiments. Numerical results also show that both stable and unstable surface kink solitons exist. However, all the surface bubble solitons are unstable.

Derivation and analysis of a phase field crystal model for a mixture of active and passive particles

We discuss an active phase field crystal (PFC) model that describes a mixture of active and passive particles. First, a microscopic derivation from dynamical density functional theory is presented that includes a systematic treatment of the relevant orientational degrees of freedom. Of particular interest is the construction of the nonlinear and coupling terms. This allows for interesting insights into the microscopic justification of phenomenological constructions used in PFC models for active particles and mixtures, the approximations required for obtaining them, and possible generalizations. Second, the derived model is investigated using linear stability analysis and nonlinear methods. It is found that the model allows for a rich nonlinear behavior with states ranging from steady periodic and localized states to various time-periodic states. The latter include standing, traveling, and modulated waves corresponding to spatially periodic and localized traveling, wiggling, and alternating peak patterns and their combinations.

Bifurcation analysis based on new macro two-velocity difference model

Based on the new macroscopic two-velocity difference model, this paper analyzes the linear stability of the new model and studies the nonlinear bifurcation theory. First, the linear stability analysis method is used to study the stability conditions of the shock wave in the model. Then, considering the long wave model in the coarse-grained scale, the reduced perturbation method is used to analyze the characteristics of the traffic flow in the metastable region, and the solitary wave solution of the Korteweg-de Vries (KdV) equation in the metastable region is derived. In addition, by using the bifurcation analysis method, the type, and stability of the equilibrium solution are discussed and the existing conditions of the saddle-node bifurcation are proven. Then, taking the saddle-node bifurcation as the starting point, we draw the density space-time diagram and phase plane diagram of the system. It is proven that the newly proposed model can describe complex traffic phenomena such as stop-and-go and sudden changes in stability, which is of great help to solve traffic congestion.