Phase Plane Analysis Method Research Articles

Existence of traveling wave solutions for density-dependent diffusion competitive systems

In this paper we are concerned with the existence of traveling wave solutions for two species competitive systems with density-dependent diffusion. Since the density-dependent diffusion is a kind of nonlinear diffusion and degenerates at the origin, the methods for proving the existence of traveling wave solutions for competitive systems with linear diffusion are invalid. To overcome the degeneracy of diffusion, we construct a nonlinear invariant region Ω near the origin. Then by using the method of phase plane analysis, we prove the existence of traveling wave solutions connecting the origin and the unique coexistence state, when the speed c is large than some positive value. In addition, when one species is density-dependent diffusive while the other is linear diffusive, via the change of variables and the central manifold theorem, we prove the existence of the minimal speed c∗ . And for c⩾c∗ , traveling wave solutions connecting the origin and the unique coexistence state still exist. In particular, when c=c∗ , we find that one component of the traveling wave solution is sharp type while the other component is smooth, which is a different phenomenon from linear diffusive systems and scalar equations.

Asymptotic Stability of Solutions to a Hyperbolic-Elliptic Coupled System of Radiating Gas on the Half Line

This paper is concerned with the asymptotic stability of the solution to an initial-boundary value problem on the half line for a hyperbolic-elliptic coupled system of radiating gas, where the data on the boundary and at the far field state are defined as $u_-$ and $u_+$ satisfying $u_-<u_+$. For the scalar viscous conservation law case, it is known by the work of Liu, Matsumura, and Nishihara [SIAM J. Math. Anal., 29 (1998), pp. 293--308] that the solution tends toward a rarefaction wave or stationary solution or superposition of these two kinds of waves depending on the distribution of $u_\pm$. Motivated by their work, we prove the stability of the above three types of wave patterns for the hyperbolic-elliptic coupled system of radiating gas with small perturbation. A singular phase plane analysis method is introduced to show the existence and the precise asymptotic behavior of the stationary solution, especially for the degenerate case: $u_-<u_+=0$ such that the system has inevitable singularities. The stability of the rarefaction wave, the stationary solution, and their superposition is proved by applying the standard $L^2$-energy method.

Multibody system dynamic analysis and payload swing control of tower crane

A new payload swing control method considering the vibration of tower crane is proposed in this study. The coupling between the structural vibration and the payload swing is neglected in existing studies on tower crane swing suppression. Changes of sling length are considered as disturbances. They cause the existing methods of swing suppression method ineffectual in the actual working situation. In contrast, the structural vibration of the tower crane is taken into account in this study. At the same time, the sling length is regarded as a control variable to reduce the payload swing. A new swing suppression algorithm is proposed in conjunction with phase plane analysis method. In addition, a systematical tower crane multibody system dynamic analysis platform with changing of sling length is established to verify the effectiveness of the algorithm. The traditional finite element method is used to model the tower body and boom of tower crane while the Arbitrary Lagrange Euler Absolute Nodal Coordinate Formulation (ALE-ANCF) cable element is applied for modeling the sling. This two parts are connected with sliding joint obtaining the tower crane multibody system model. Numerical examples show that the proposed method can reduce the swing amplitude the payload effectively when considering the coupling between the structural vibration and the payload swing.

Study on nonlinear dynamic behavior and stability of aviation pressure servo valve-controlled cylinder system

In this paper, aiming at the phenomenon of self-excited oscillation caused by nonlinear factors in a large aircraft wheel brake control system, the nonlinear dynamic behavior of the pressure servo valve-controlled cylinder system (PSVCS) is studied, and the influence law of the key parameters on the nonlinear self-excitation behavior is obtained. On this basis, the stability of the PSVCS is analyzed both in time domain and frequency domain, and it is proved in principle that the PSVCS is a stable self-closed-loop control system. Firstly, the nonlinear dynamics model of the PSVCS is established in this paper. Secondly, using the method of phase plane analysis, the nonlinear dynamic behavior of the PSVCS and the influence law of key parameters on the system are studied. Thirdly, the nonlinear system of the PSVCS is transformed into a segmented local linear system, and the stability of the prestage and the power stage is analyzed, respectively. Finally, through a performance test platform, which is used to simulate the load of PSVCS, the theoretical analysis results of this paper are verified experimentally under different working conditions. The final experimental results show that both the nonlinear dynamic model established in this paper and the influence law of the key parameters obtained by the phase plane analysis on the nonlinear self-excited oscillation behavior are correct, and the relevant conclusions can provide a reference for the design of the braking system control system.

Stability analysis and simulation based on an improved Aw-Rascle model

In this paper, the improved Aw-Rascle model with viscosity term is developed to a new nonlinear dynamical system composed of ordinary differential equations, by using traveling wave substitution and a kind of variable substitution. The stability of the system is discussed by the method of phase plane analysis after getting the travelling-wave solutions, then the final data is simulated by Matlab to verify the analysis conclusion.

Equilibrium Point analysis of macro traffic flow model considering the influence of front and rear lights

In this paper, the traveling wave solution of the macro continuum model considering the automobile tail lamp effect is obtained by using the phase plane analysis method, the type of equilibrium point of the model is determined, and the stability of the system at the equilibrium point is analyzed. This method can describe and predict the nonlinear traffic phenomena on Expressway from the perspective of system global stability. According to the theory of differential equation, the model is further analyzed to judge the type and stability of equilibrium point. Finally, the simulation diagram is used for numerical simulation. Through numerical verification, it is concluded that the numerical results are consistent with the theoretical analysis.

Analysis of equilibrium point based on nonlinear traffic flow model

In this paper, the traveling wave solution of Kerner konhauser model is obtained by phase plane analysis method, and a new model suitable for stability analysis is obtained by variable transformation. The new model can describe and predict the nonlinear traffic phenomenon on Expressway from the perspective of system global stability. In this paper, the nonlinear system and linear system are obtained respectively by traveling wave substitution and Taylor expansion at the equilibrium point. According to the qualitative theory of differential equations, the equilibrium point type and stability of the linear system are determined. Finally, the simulation results show the consistency between the numerical results and the theoretical analysis.

Analysis of equilibrium point based on viscous traffic flow model

In this paper, the phase plane analysis method is used, which can describe and predict the nonlinear traffic phenomenon on Expressway from the perspective of system global stability. The nonlinear system of the model is obtained by the traveling wave substitution and Taylor expansion of the model, and the equilibrium point of the model is solved by specifying the model parameters. According to the definite theory of differential equation, the model is further analyzed to judge the type and stability of equilibrium point. Finally, numerical verification is carried out according to the simulation diagram. Through numerical verification, the simulation results of the model are consistent with the conclusions of theoretical analysis. It can more clearly describe the change of density or speed with time or road section.

Phase plane analysis of traffic flow based on driver’s expected response model

At present, researchers have proposed many traffic flow models and research methods of traffic phenomena, but few of them are analyzed from the perspective of system stability. Therefore, this paper proposes a phase plane analysis method from the perspective of traffic flow stability. This method can describe the nonlinear traffic flow phenomenon on the road from the perspective of system global stability. The nonlinear system of the model is obtained by traveling wave substitution and Taylor expansion, and the equilibrium point of the model is solved by specifying the model parameters. According to the qualitative theory of differential equation, the model is further analyzed to judge the type and stability of equilibrium point. Finally, the simulation diagram is used for numerical verification.

Periodic bouncing solutions for Hill's type sub-linear oscillators with obstacles

<p style='text-indent:20px;'>In this paper, we prove the existence and multiplicity of subharmonic bouncing motions for a Hill's type sublinear oscillator with an obstacle. Furthermore, we also consider the existence, multiplicity and dense distribution of symmetric periodic bouncing solutions when the weight function is even. Based on an appropriate coordinate transformation and the method of phase-plane analysis, we can study our main results via Poincar<inline-formula><tex-math id="M1">\begin{document}$ \acute{e} $\end{document}</tex-math></inline-formula> map by applying some suitable fixed point theorems.

Tracking differentiator based on improved Gaussian distribution

Concerning the problems of the existing tracking differentiator, such as large computation, complex parameter tuning and output chattering, a nonlinear tracking differentiator (GTD) based on the improved Gaussian distribution was proposed. It is a simple calculating nonlinear tracking differentiator with easy parameters tuning and strong ability of noise restraint. Firstly, the amplitude gain and exponential factor are introduced to improve the Gaussian distribution, then the acceleration function is constructed by utilizing the improved Gaussian distribution. Secondly, the global asymptotic stability of the designed TD is proved by constructing Lyapunov function. And its singularity type and phase trajectory distribution are analyzed by the phase plane analysis method, and the qualitative law of parameter adjustment is also given. Finally, simulations are performed and results are compared with linear differentiator and anti-hyperbolic sinusoidal based TD. It concludes that the differential signal of the improved Gaussian-distribution-based nonlinear tracking differentiator does not have the high frequency flutter behavior in steady state. Moreover, the GTD not only has good dynamic response but also has strong filtering ability.

Solutions with concentration and cavitation to the Riemann problem for the isentropic relativistic Euler system for the extended Chaplygin gas

Abstract The solutions to the Riemann problem for the isentropic relativistic Euler system for the extended Chaplygin gas are constructed for all kinds of situations by using the method of phase plane analysis. The asymptotic limits of solutions to the Riemann problem for the relativistic extended Chaplygin Euler system are investigated in detail when the pressure given by the equation of state of extended Chaplygin gas becomes that of the pressureless gas. During the process of vanishing pressure, the phenomenon of concentration can be identified and analyzed when the two-shock Riemann solution tends to a delta shock wave solution as well as the phenomenon of cavitation also being captured and observed when the two-rarefaction-wave Riemann solution tends to a two-contact-discontinuity solution with a vacuum state between them.

Study on the effect of the motion state of interface on the damping characteristics of SSI system

Based on the theory of discontinuous dynamical systems, this paper investigates the effect of the motion state of interface on the damping characteristics of SSI system through the phase plane analysis method. The necessary and sufficient conditions for the coordinated motion state of interface between soil and structure are obtained. The relationship between the motion state of interface and the damping system is investigated via the shaking table test. It is verified that the coordinated motion state of interface is the decisive factor for the identification of damping system. The phase trajectories of the interface and the transfer functions of the system after the different magnitudes of vibration are further given. The test results show that the motion coordination mechanism is developed between soil and structure. The coordination mechanism enables soil and structure to have a unified and collaborative motion state after a small magnitude of the vibration. The SSI system can be viewed as an approximately classical damping system under a certain dynamic excitation.

The Riemann problem and the limit solutions as magnetic field vanishes to magnetogasdynamics for generalized Chaplygin gas

This paper is concerned with the Euler equations in the magnetogasdynamics for generalized Chaplygin gas. The global solutions to the Riemann problems of the Euler equations in the magnetogasdynamics for generalized Chaplygin gas are obtained constructively by using phase plane analysis method. The formation of delta shock wave is studied as magnetic field vanishes. The limit behaviors of the Riemann solutions as magnetic field vanishes are also obtained.

Discontinuous traveling wave entropy solutions for a sedimentation–consolidation model

This paper deals with a mathematical model of gravitational sedimentation–consolidation processes, for which the existence of discontinuous traveling wave entropy solutions is established by using the phase plane analysis method.

Discontinuous traveling waves for scalar hyperbolic-parabolic balance law

This paper is concerned with the existence of traveling waves for the scalar hyperbolic-parabolic balance law. Using a phase-plane analysis method, we first prove the existence of an increasing traveling wave solution in $C^{1}(\mathbb{R})$ . Then we construct a family of discontinuous periodic traveling wave entropy solutions.

Two-dimensional Riemann problem involving three contact discontinuities for [formula omitted] hyperbolic conservation laws in anisotropic media

In this study, we consider the two-dimensional Riemann problem for a system of conservation laws, which models polymer flooding in an anisotropic oil reservoir. The initial data are constants in three sectors centered at the origin, which only involve three contact discontinuities. By the generalized characteristic analysis method and the phase plane analysis method, we find that the solutions are not unique for certain values of the initial data. We propose an additional stability condition for the interaction of the contact discontinuities. By this stability condition, all of the exact solutions and their corresponding criteria can be obtained. The solutions exhibit various geometrical structures. In particular, envelope rarefaction waves develop in some solutions.

An improved method for evaluating the rotational speed stability of a hydro-viscous clutch in mixed lubrication

Abstract Rotational speed stability is an important evaluation indicator of the performance of a hydro-viscous clutch (HVC). To improve the rotational speed stability of HVCs in mixed lubrication and the running condition of the friction pairs, the speed stability of an HVC in mixed lubrication was studied. To this end, the friction coefficients of both copper-based and paper-based friction pairs were experimentally tested using an MM1000-III wet friction machine. Theoretically, a torsional vibration model of the system is presented. The phase plane analysis method is applied to evaluate the stability of the torsional vibration model, where a critical negative gradient (CNG) is defined. The results show that the friction coefficient in mixed lubrication is an important parameter for the stability of the rotational speed. The system will be unstable when the negative gradient of the friction coefficient-slip speed is larger than the CNG. According to the definition of the CNG, suggestions regarding choice of friction pairs are made to improve the rotational speed stability of an HVC in mixed lubrication.

Phase Plane Analysis Method of Nonlinear Traffic Phenomena

A new phase plane analysis method for analyzing the complex nonlinear traffic phenomena is presented in this paper. This method makes use of variable substitution to transform a traditional traffic flow model into a new model which is suitable for the analysis in phase plane. According to the new model, various traffic phenomena, such as the well-known shock waves, rarefaction waves, and stop-and-go waves, are analyzed in the phase plane. From the phase plane diagrams, we can see the relationship between traffic jams and system instability. So the problem of traffic flow could be converted into that of system stability. The results show that the traffic phenomena described by the new method is consistent with that described by traditional methods. Moreover, the phase plane analysis highlights the unstable traffic phenomena we are chiefly concerned about and describes the variation of density or velocity with time or sections more clearly.

A Landesman-Lazer type condition for second-order differential equations with a singularity at resonance

In this paper, we are concerned with the existence of positive periodic solutions for second-order differential equations with a singularity at resonance. A Landesman-Lazer type condition is given to obtain the existence of positive periodic solutions using phase-plane analysis method and topological degree method.