Stability Of Equilibrium Solutions Research Articles

Multifractal chaotic attractors in a system of delay-differential equations modeling road traffic.

We study a system of delay-differential equations modeling single-lane road traffic. The cars move in a closed circuit and the system's variables are each car's velocity and the distance to the car ahead. For low and high values of traffic density the system has a stable equilibrium solution, corresponding to the uniform flow. Gradually decreasing the density from high to intermediate values we observe a sequence of supercritical Hopf bifurcations forming multistable limit cycles, corresponding to flow regimes with periodically moving traffic jams. Using an asymptotic technique we find approximately small limit cycles born at Hopf bifurcations and numerically preform their global continuations with decreasing density. For sufficiently large delay the system passes to chaos following the Ruelle-Takens-Newhouse scenario (limit cycles-two-tori-three-tori-chaotic attractors). We find that chaotic and nonchaotic attractors coexist for the same parameter values and that chaotic attractors have a broad multifractal spectrum. (c) 2002 American Institute of Physics.

Asymptotic Behavior of Solutions to the Cahn-Hilliard Equation in Spherically Symmetric Domains

The Cahn-Hilliard equation is a fourth-order parabolic partial differential equation that is one of the leading models for the study of phase separation in isothermal, isotropic, binary mixtures, such as molten alloys. The asymptotic behavior of solutions to the Cahn-Hilliard equation with Dirichlet boundary conditions and the associated stationary problem have been studied. In particular, it is proved that the only possible stable equilibrium solutions in spherically symmetric domains are spherically symmetric and monotone in the radial direction.

Slow travelling waves on a finite interval for Burgers’-type equations

In this paper we study for small positive ε the slow motion of the solution for evolution equations of Burgers’ type with small diffusion, () u t=εu xx+F(u) x, u(x,0)=u 0(x), u(±1,t)=u ± on the bounded spatial domain [−1,1]; F is a smooth nonpositive function having only a finite number of zeros (at least two) between u − and u +, all of finite order. The initial and boundary value problem (★) has a unique asymptotically stable equilibrium solution that attracts all solutions starting with continuous initial data u 0. On an interval [−1−c 0ε,1+c 0ε], c 0>0 the differential equation has slow speed travelling wave solutions generated by profiles that satisfy the boundary conditions of (★). During a long but finite time interval, such travelling waves suitably describe the slow long-term behaviour of the solution of evolution problem (★). Their speed characterizes the local velocity of the slow motion with algebraic precision (w.r.t. ε) in general, and with exponential precision, if F has only two zeros of first order located at u + and u −. A solution that starts near a travelling wave, moves in a small neighbourhood of such a travelling wave during a long time interval (0, T). If F has zeros of order higher than 1, the equilibrium and the travelling wave are multi-shock solutions of (★). This situation differs strongly from the case where F has only a first-order zero at both u ±, studied by the authors in a previous paper. In this paper we consider multi-shock solutions of (★). Moreover, we improve some results of the previous paper, allowing a larger ball of initial data.

Simulation of Second Moments of Fluctuations of the Velocity and Temperature of Particles in Equilibrium Turbulent Flows

An analysis is performed of equilibrium solutions of equations for the second moments of fluctuations of the velocity and temperature of particles in a homogeneous shear flow, in a homogeneous flow subjected to tensile or compressive strain, and in a wall layer. The stability of equilibrium solutions to small perturbations is investigated. Algebraic models are given for turbulent stresses in the dispersed phase.

Lotka–Volterra models and the global vegetation pattern

How can such a classic object of mathematical ecology as the Lotka–Volterra competition model for a community of n competing species be used for description of the global vegetation dynamics under the climate change? We assume that the present spatial distribution of global vegetation is a stable equilibrium solution of corresponding Lotka–Volterra equations. The problem is how to construct some discrete structure on a continuous set of parameters, in this context remaining in a framework of a continuous model description? The problem can be solved if a dynamical system with multiple equilibria is considered. In this case, a continuous quasi-stationary change of parameters induces a jump from one to another equilibrium, since they have different stability domain in the parametric space. On this conceptual base a special class of Lotka–Volterra equations is developed and a biologically interpreted procedure for estimating their coefficients is suggested. For this we must have maps of the annual production and biomass, a life span of each vegetation type, and a map of the current geographical distribution of different vegetation, i.e. the current global vegetation pattern (GVP). The latter is needed for the construction of the ‘elementary map’, which is an important part of the model. Another important part is the formula, which describes an annual production dependence on the temperature and precipitation (for instance, Lieth's formula can be used). Considering a one-dimensional particular case for n=2 we have the analytical formulas describing a shift of borders between two vegetation zones under the climate change. It was shown that two different types of the transition zones, namely, ‘soft’ and ‘hard’, exist in this case. If the soft zone is characterised by the continuous and smooth replacement of one type of vegetation by another, then the hard zone is a typical ‘fractal’ structure with a mosaic of different vegetation. A special calibration procedure for the Lotka–Volterra model describing the dynamics of one-dimensional GVP with n types of vegetation is suggested. It is based on the stability theorem for a system with multiple equilibria and a special averaging method applying to real geographical distributions of the annual production and the biomasses. The results are used for estimating the shift of one transition zone between taiga and steppe in the Central Siberia under the climate change (CO 2-doubling scenario).

Static path jumping to attain postbuckling equilibria of a compressed circular cylinder

The postbuckling behavior of an axially compressed circular cylindrical shell is exceedingly complicated due to an infinite number of closely spaced postbuckling branches and bifurcation points. The minimum strength existing in the deep bottom of the postbuckling region may serve as a design limit. The primary concern in this present paper is to compute this stable postbuckling equilibrium solution by two different approaches: One is to repeat the procedures of tracing unstable branches, pinpointing bifurcation points and branch-switching in order to carefully approach to the target. The other is to trigger a static jump to the target by two-parametric loading. As a numerical example, a perfect circular cylindrical panel is analyzed to show that a direct jump from the undeformed state to a stable postbuckling solution is possible with a proper choice of the load perturbation.

Simple Model of the Plasma Torch Developed for Application to an Arc-Plasma Waste-Treatment System

A simple theoretical model describing physics of the plasma torch plume is developed in connection with its applications to the arc-plasma waste-treatment system. The theoretical analysis is carried out by making use of Bernoulli's pressure-balance equation, which provides a stable equilibrium solution of the gas density in the plume ejected from the torch into a high-pressure reactor chamber with 4ε < 1. The pressure depression parameter ε is proportional to the gas temperature and inversely proportional to the square of the chamber pressure. In a low-pressure chamber, characterized by 4ε >1, there is no stable equilibrium solution satisfying Bernoulli's equation. Therefore, it is expected that the observable plasma data may change abruptly as the chamber pressure crosses the borderline defined by 4ε = 1. Indeed, most of the plasma data measured in an experiment change abruptly at the pressure borderline of 4ε = 1. The oxygen torch plume is theoretically analyzed for the arc-plasma waste-treatment system. Due to a very high arc-plasma temperature inside the torch, most of the oxygen molecules are atomized when they exit the torch and enter the reactor chamber. The oxidation process of materials by oxygen atoms can be easily achieved at a reaction rate of a million limes faster than that by oxygen molecules. It is therefore essential to evaluate the oxygen atom concentration in the oxygen plume propagating through the waste-treatment system. A numerical example indicates that a high concentration of oxygen atoms exists even at a considerable distance (i.e., 40 cm) from the torch exit in typical torch operation conditions.

A Study on the Critical Speed of Railway Vehicles

This research has been performed to reveal the hysteresis phenomena of the hunting motion in a railway passenger car having a bolster. Since linear analysis can not explain them, bifurcation analysis is used to predict its outbreak velocities in this paper. However bifurcation analysis is attended with huge computing time, thus this research proposes more effective numerical algorithm to reduce it than previous researches. Stability of periodic solution is obtained by adapting of Floquet theory while stability of equilibrium solutions is obtained by eigen-value analysis. As a result, linear and nonlinear critical speed are acquired. Full scale roller rig test is carried out for the validation of the numerical result. Finally, it is certified that there are many similarities between numerical and test results.

Modeling rotational effects in eddy-viscosity closures

A new approach to sensitize turbulence closures based on the linear eddy-viscosity hypothesis to rotational effects is proposed. The principal idea is to `mimic' the behavior of a second moment closure (SMC) in rotating homogeneous shear flow; depending on the ratio of the mean flow to the imposed rotational time scales, the model should be able to bifurcate between two stable equilibrium solutions. These solutions correspond to exponential or algebraic time dependent growth or decay of turbulent kinetic energy. This fundamental behavior of SMCs is believed to be of importance also in the prediction of non-equilibrium turbulence. A near-wall turbulence model which is based on the linear eddy-viscosity hypothesis is modified in the present study. Wall proximity effects are modeled by the elliptic relaxation approach. This closure has been successfully applied in the computation of complex, non-equilibrium flows in inertial frames of reference. The objective of the present study is to extend the predictive capability of the model to include flows dominated by rotational effects. The new model is calibrated in rotating homogeneous turbulent shear flow and subsequently tested in three different cases characterized by profound effects of system rotation or streamline curvature. It is able to capture many of the effects due to imposed body forces that the original closure is incapable of. Good agreement is obtained between the present predictions and available experimental and DNS data.

Double Curvature Bending of Variable-Arc-Length Elasticas

This paper deals with the double curvature bending of variable arc-length elasticas under two applied moments at fixed support locations. One end of the elastica is held while the other end portion of the elastica may slide freely on a frictionless support at a prescribed distance from the held end. Thus, the variable deformed length of the elastica between the end support and the frictionless support depends on the relative magnitude of the applied moments. To solve this difficult and highly nonlinear problem, two approaches have been used. In the first approach, the elliptic integrals are formulated based on the governing nonlinear equation of the problem. The pertinent equations obtained from applying the boundary conditions are then solved iteratively for solution. In the second approach, the shooting-optimization method is employed in which the set of governing differential equations is numerically integrated using the Runge-Kutta algorithm and the error norm of the terminal boundary conditions is minimized using a direct optimization technique. Both methods furnish almost the same stable and unstable equilibrium solutions. An interesting feature of this kind of bending problem is that the elastica can form a single loop or snap-back bending for some cases of the unstable equilibrium configuration.

Stabilization of Vortices in the Ginzburg--Landau Equation with a Variable Diffusion Coefficient

We study equilibria of the Ginzburg--Landau equation with a variable diffusion coefficient on a bounded planar domain subject to the Neumann boundary condition. It has been previously shown that if the diffusion coefficient is constant and the ambient domain is convex, the system does not carry stable vortices in the sense that any stable equilibrium solution is a constant of modulus 1. In this article we shall prove that arbitrarily given a domain, an appropriate choice of inhomogeneous diffusion coefficient yields a stable equilibrium solution having vortices. We can even manage to make the configuration of stable vortices close to prescribed locations. Our method is to minimize the free energy functional in suitably constructed positive invariant regions for the time-dependent Ginzburg--Landau equation.

Equilibrium Analysis of a Mathematical Model for the Spread of AIDS in a Two Sex Population with Mixing Constraints

For a mathematical model for the spread of HIV by sexual transmission in a heterosexual population we analyse the existence and stability of equilibrium solutions. The model is designed to investigate the effects of a fundamental constraint in any social/sexual mixing process for heterogeneous populations. The group contact constraint conserves the number of new sexual partnerships formed per unit time between the sexes, and will have at least a quantitative influence on the dynamics of the model. The analysis has been carried out for general and specific forms of the sexual activity rates (the mean number of sexual partners per unit time for a typical individual). In general we define a threshold parameter R0, the Reproductive Number, which is a key determinant of the behaviour of the model. We show that in general if R0&gt;1 there is at least one endemic equilibrium. The specific cases of a dominance sexual activity rate and proportional sexual activity rates are discussed in more detail. Multiple endemic equilibria can occur for the former but not the latter.

Neutron stars in scalar-tensor theories of gravity and catastrophe theory

We investigate neutron stars in scalar-tensor theories. We examine their secular stability against spherically symmetric perturbations by use of a turning point method. For some choices of the coupling function contained in the theories, the number of the stable equilibrium solutions changes and the realized equilibrium may change discontinuously as the asymptotic value of the scalar field or total baryon number is changed continuously. The behaviour of the stable equilibrium solutions is explained by fold and cusp catastrophes. Whether the cusp catastrophe appears or not depends on the choices of the coupling function. These types of the catastrophes are structurally stable. Recently discovered spontaneous scalarization, which is non-perturbative strong-field phenomenon due to the presence of the gravitational scalar field, is well described in terms of the cusp catastrophe.

Stability properties of the Boussinesq equations

We consider the stability of equilibrium solutions of the Boussinesq equations in an infinite layer \(R^2\times(-{1\over2},{1\over2})\) heated from below. The equilibria are assumed to be space-periodic in the plane direction. Whereas their situation is different for non-periodic disturbance has been studied extensively the situation is different for non-periodic perturbations. Our interest here is focused on disturbances which are square-integrable in the plane variables. For the motionless state and convection rolls as equilibrium solutions their stability against perturbations in \(L^2(R^2\times(-{1\over2},{1\over2}))\) is compared with their stability against periodic ones.

GLOPEQ: A new computational tool for the phase and chemical equilibrium problem

Calculation of phase and chemical equilibrium represents a crucial phase in the modeling of many separation processes. For conditions of constant temperature and pressure, a necessary and sufficient condition for the true equilibrium solution is that (i) the total Gibbs free energy of the system be at its global minimum, or (ii) the minimum of the tangent plane distance function be nonnegative for all phase models used to represent the system. In this work, the goal is to obtain equilibrium solutions corresponding to a global minimum of the Gibbs free energy as efficiently as possible, for cases where the liquid phase or phases can be modeled by the NRTL, UNIQUAC, UNIFAC, Wilson, modified Wilson and ASOG equations. Vapor phases whose behavior can be described as ideal can also be handled. In achieving this goal, there are two distinct problems of relevance: (i) the minimization of the Gibbs free energy, denoted (G), and (ii) the minimization of the tangent plane distance function, or the tangent plane stability criterion, denoted (S). For all these activity coefficient models, GLOPEQ (GLobal OPtimization for the Phase and chemical EQuilibrium problem) can guarantee global solutions for problems (G) and (S), but a combined algorithm employs them in tandem, using (G) to generate candidate equilibrium solutions which can then be verified for thermodynamic stability by solving (S). Two key features of the combined algorithm are that (i) as much information as is possible is obtained from local searches, and (ii) it is preferable to verify a globally stable equilibrium solution using the tangent plane criterion, as this problem contains fewer variables than the minimization of the Gibbs free energy. Results for several examples are presented, and all but one of them are for the case of phase equilibrium, due to the paucity of examples for reacting systems that employ excess Gibbs free energy models.

The propagation of premixed flames in closed tubes

A nonlinear evolution equation that describes the propagation of a premixed flame in a closed tube has been derived from the general conservation equations. What distinguishes it from other similar equations is a memory term whose origin is in the vorticity production at the flame front. The two important parameters in this equation are the tube's aspect ratio and the Markstein parameter. A linear stability analysis indicates that when the Markstein parameter α is above a critical value αc the planar flame is the stable equilibrium solution. For α below αc the planar flame is no longer stable and there is a band of growing modes. Numerical solutions of the full nonlinear equation confirm this conclusion. Starting with random initial conditions the results indicate that, after a short transient, a at flame develops when α&gt;αc and it remains flat until it reaches the end of the tube. When α&lt;αc, on the other hand, stable curved flames may develop down the tube. Depending on the initial conditions the flame assumes either a cellular structure, characterized by a finite number of cells convex towards the unburned gas, or a tulip shape characterized by a sharp indentation at the centre of the tube pointing toward the burned gases. In particular, if the initial conditions are chosen so as to simulate the elongated finger-like flame that evolves from an ignition source, a tulip flame evolves downstream. In accord with experimental observations the tulip shape forms only after the flame has travelled a certain distance down the tube, it does not form in short tubes and its formation depends on the mixture composition. While the initial deformation of the flame front is a direct result of the hydrodynamic instability, the actual formation of the tulip flame results from the vortical motion created in the burned gas which is a consequence of the vorticity produced at the flame front.

Evolution of Domains of Attraction of a Forced Beam with Two-Mode Interaction

When a dissipative dynamical system has multiple attractors, it is a task to determine and recognize the global domain of attraction of each attractor. In this paper we study the global behavior of a forced hinged-clamped beam with two-mode interaction. The equation of motion of the beam is reduced to four first-order autonomous ordinary differential equations. The system has an internal resonance condition of ω2 ≈ 3ω1, where ω1 and ω2 denote natural frequencies of the first and second modes, respectively. When the excitation frequency is near ω1, the system can have three equilibrium solutions, among which two are asymptotically stable and one is unstable. We examine how the domains of attraction of two stable equilibrium solutions evolve as the forcing frequency is varied across jump points. By using a special plane which contains all equilibrium points, called the principal plane, the global domains of attraction can be discussed more effectively. Results show that knowledge of this evolution helps us better understand the jump phenomenon.

Population Dynamics with Sequential Density-Dependencies

We analyse the importance of sequential density-dependent processes to population dynamics of single species. We divide a year into several processes of density-dependent reproduction and/or mortality. A sequence of n processes can be arranged in n ! different sequences. However, only (n - 1) ! of these represent unique relative orderings that have different stability properties and dynamics. Models with several sequential density-dependent processes have a much wider repertoire of dynamics than, e.g., ordinary models based on the logistic equation. Stable equilibrium density and the maximum density of cycles and unstable dynamics do not necessarily increase with increasing b (maximum per capita birth rate). The maps of density at time t + 1 (x t+1 ) versus density at time t (x t ) can have more than one hump, i.e., be bi- or multimodal, with multiple equilibria. In this type of system, chaos is not the only inevitable outcome of increased b. Instead stable equilibrium and/or periodic solutions may occur beyond the chaotic region as b increases. It is suggested that this type of model may apply to many kinds of organisms in seasonal environments. The explicit consideration of sequential density-dependence may be of critical importance for resource and conservation managers, to avoid switches between multiple equilibria or extinction due to poorly timed harvest or pest control.

On equilibrium solutions of diffusion equations with nonlinear boundary conditions

The existence or the nonexistence of nonconstant stable equilibrium solutions for a diffusion equation with nonlinear Neumann boundary conditions is studied. We prove the nonexistence of nonconstant stable equilibria when the nonlinearity has a small Lipschitz constant or a second derivative of constant sign or either when the domain is a ball. We construct an example of existence for a connected domain with several disconnected boundary components.

On the solution of the model of two co-affected species

This paper discusses a model of two coeffected species with diffusion system, where the birth functions have the characteristic index decline, and then the circumstance factors α1 which are connected with natural resources, and the birth parameters βi which could be controlled are introduced. By means of the upper-lower solutions, the existence, uniqueness and the local stability of equilibrium solution of the model are discussed. It’s discovered that the birth parameters βi determine the developing tendency when other parameters are comparatively stable.