The existence of traveling wave front solutions of a model of the Belousov-Zhabotinskii chemical reaction

The boundary value problem $\text{?u}\text{?t′}\text{= u(1 ? u ? rv) +}\text{?}{}^2\text{u}\text{?x′}{}^2$$\text{?v}\text{?t′}\text{= ? buv +}\text{?}{}^2\text{y}\text{?x′}{}^2$ where u(? ∞, t′) = v(∞, t′) = 0 andv(? ∞, t′) = u(∞, t′) = 1 for each t′ > 0 has been proposed by Murray as a model for the Belousov-Zhabotinskii chemical reaction. Here u and v are proportional to the concentrations of bromous acid and bromide ion, respectively. We prove that there is a range of values for b and r over which the boundary value problem has traveling wave front solutions.     

Topological analysis of chaotic time series data from the Belousov-Zhabotinskii reaction

Summary We have applied topological methods to analyze chaotic time series data from the Belousov-Zhabotinskii reaction. First, the periodic orbits shadowed by the data set were identified. Next, a three-dimensional embedding without self-intersections was constructed from the data set. The topological structure of that flow was visualized by constructing a branched manifold such that every periodic orbit in the flow could be held by the branched manifold. The branched manifold, or induced template, was computed using the three lowest-period orbits. The organization of the higher-period orbits predicted by this induced template was compared with the organization of the orbits reconstructed from the data set with excellent results. The consequences of the presence of certain knots found in the data are discussed.     

The asymptotic behavior of solutions of a system of reaction-diffusion equations which models the Belousov-Zhabotinskii chemical reaction

We investigate the boundary value problem $\text{?u}\text{?t}\text{=}\text{?}{}^2\text{u}\text{?x}{}^2\text{+ u(1 ? u ? rv)}$, $\text{?v}\text{?t}\text{=}\text{?}{}^2\text{v}\text{?x}{}^2\text{? buv}$, u(?∞, t) = v(∞, t) = 0, u(∞, t) = 1, and v(?∞, t) = γ ?t > 0 where r > 0, b > 0, γ > 0 and x?R. This system has been proposed by Murray as a model for the propagation of wave fronts of chemical activity in the Belousov-Zhabotinskii chemical reaction. Here u and v are proportional to the concentrations of bromous acid and bromide ion, respectively. We determine the global stability of the constant solution (u, v) ≡ (1,0). Furthermore we introduce a moving coordinate and for each fixed x?R we investigate the asymptotic behavior of u(x + ct, t) and v(x + ct, t) as t → ∞ for both large and small values of the wave speed c ? 0.     

A threshold phenomenon for embeddings of $${H^m_0}$$ into Orlicz spaces

Given an open bounded domain \({\Omega\subset\mathbb {R}^{2m}}\) with smooth boundary, we consider a sequence \({(u_k)_{k\in\mathbb{N}}}\) of positive smooth solutions to

$\left\{\begin{array}{ll} (-\Delta)^m u_k=\lambda_k u_k e^{mu_k^2} \quad\quad\quad\quad\quad {\rm in}\,\Omega\\ u_k=\partial_\nu u_k=\cdots =\partial_\nu^{m-1} u_k=0 \quad {\rm on }\, \partial \Omega, \end{array}\right.$

where λ _k → 0⁺. Assuming that the sequence is bounded in \({H^m_0(\Omega)}\) , we study its blow-up behavior. We show that if the sequence is not precompact, then

$\liminf_{k\to\infty}\|u_k\|^2_{H^m_0}:=\liminf_{k\to\infty}\int\limits_\Omega u_k(-\Delta)^m u_k dx\geq \Lambda_1,$

where Λ₁ = (2m ? 1)!vol(S ^2m ) is the total Q-curvature of S ^2m .

     

System catastrophe: A distributive model for collective phenomenon

The phenomenon of system catastrophe often occurs in a system with a network structure. A system's resources can be utilized in two different modes: efficiently or inefficiently. When actions with inefficient mode pose no threat to other users or, in other words, when they employ resources that would otherwise be idle, they do not waste the system's resources at all. But when critical levels of inefficient uses of system's resources are reached, there is a sudden decrease in the capacity of the system due to the multiplication effects of inefficient factors. This collective inefficiency results in everyone getting worse in average. The common theme behind the catastrophe phenomenon demonstrates a possible explanation for the famous question about the choice between market and hierarchy. That is, when all firms pursue their own individual interests, resulting in a collective breakdown, they turn to consolidated ways of carrying out transactions.     

Persistent oscillations in a threshold adjustment model

In this paper we present a simple time-continuous behavioural model of habit formation. Addictive behaviour is damped by a threshold which adapts itself to the habit. This adaptive behaviour of the threshold may lead to periodic fluctuations of the consumption rate, the habit and the threshold. It turns out that both a low adjustment rate of the threshold as well as a steep consumption function favour oscillatory patterns.     

Asymptotic Behavior of Solutions for the Belousov-Zhabotinskii Reaction Diffusion System

The present paper characterizes the asymptotic behavior of the timedependenr solution of the coupled Belousov-Zhabotinskii reaction diffusion equations in relation to the steady-state solutions ot the corresponding boundary value problem. This characterization leads to an cxplicit reltionship among the various physical constants and the boundary and initial functions.     

Asymptotic Solutions of the Field-Noyes Model for the Belousov Reaction—II. Plane Waves

The spatially homogeneous limit cycle solutions found by Stanshine and Howard [7] for the Field-Noyes model of the Belousov reaction are used to obtain plane wave solutions for the related reaction-diffusion equations. Perturbation techniques are implemented first to find long wavelength waves and then successively to find shorter wavelength waves. Plane wave solutions are then shown to exist for sufficiently short wavelengths for parametric values which are physically plausible but for which spatially homogeneous oscillations do not (apparently) exist.     

Asymptotic Solutions of the Field-Noyes Model for the Belousov Reaction—I. Homogeneous Oscillations

Multiple-time-scale techniques are used to solve the non-linear autonomous system used by Field and Noyes to model the chemical oscillations of the Belousov reaction. An asymptotic representation, valid for a wide range of parameters, is found for a spatially homogeneous limit-cycle solution. For certain values of the parameters, two limit-cycle solutions are shown (asymptotically) to exist. For parameter values for which the limit cycle appears to be unique, it is shown to be linearly stable. The asymptotic solution is shown to correspond excellently to the numerical solution calculated by Field and Noyes for one set of parameters.     

A note on the perturbed compound poisson risk model with a threshold dividend strategy

In this paper, we consider the Perturbed Compound Poisson Risk Model with a threshold dividend strategy （PCT）. Integro-differential equations （IDE） for its Cerber-Shiu functions and dividend payments function are stated. We maily focus on deriving the boundary conditions to solve these equations.     

A mathematical model of imbibition phenomenon in heterogeneous porous media during secondary oil recovery process

The present paper discuses the solution of one-dimensional mathematical model for counter-current water imbibition phenomenon occurring into an oil-saturated cylindrical heterogeneous porous matrix. During secondary oil recovery process when water is injected in oil formatted heterogeneous porous matrix then at common interface the counter current imbibition phenomenon occurs due to the difference of viscosity of water and oil which satisfies imbibition condition _Vi=-_Vn$V_i=-V_n$. The governing differential equation of this phenomenon is in the form of non-linear partial differential equation which has been converted into non-linear ordinary differential by using similarity transformation. The solution of this problem has been obtained in term of power series by using appropriate boundary condition at common interface. The graphical presentation is obtained by using MATLAB and final solution physically interpreted.     

On the renewal risk model under a threshold strategy

In this paper, we consider the renewal risk process under a threshold dividend payment strategy. For this model, the expected discounted dividend payments and the Gerber–Shiu expected discounted penalty function are investigated. Integral equations, integro-differential equations and some closed form expressions for them are derived. When the claims are exponentially distributed, it is verified that the expected penalty of the deficit at ruin is proportional to the ruin probability.     

Diffusive monotonicity and threshold dynamics of delayed reaction diffusion equations

For abstract functional differential equations and reaction-diffusion equations with delay, an exponential ordering is introduced which takes into account the spatial diffusion. The induced monotonicity of the solution semiflows is established and applied to describe the threshold dynamics (extinction or persistence/convergence to positive equilibria) for a nonlocal and delayed reaction-diffusion population model.     

A stochastic model of a chemical reaction with diffusion

Summary Stochastic systems of Brownian motions with multiple deletion of particles are introduced to model a chemical reaction with diffusion. Convergence to the solution of a deterministic nonlinear reaction-diffusion equation is proved without high-density assumptions.     

Optimality of the threshold dividend strategy for the compound Poisson model

In this paper, we consider the optimal dividend problem for the compound Poisson risk model. We assume that dividends are paid to the shareholders according to an admissible strategy with dividend rate bounded by a constant. Our objective is to find a dividend policy so as to maximize the expected discounted value of dividends until ruin. We give sufficient conditions under which the optimal strategy is of threshold type.     

A note on the Kunze-Stein phenomenon

The authors show that convolution with an L^p function k is a bounded operator on L²(G), 1 ? p < 2, for G noncompact semisimple and k having invariance properties.     

Existence of traveling-wave type solutions for the Belousov-Zhabotinskii system of equations. II

Moscow. Translated from Sibirskíi Matematicheskii Zhurnal, Vol. 32, No. 3, pp. 47–59, May–June, 1991.