Small random perturbations of a dynamical system with blow-up

We study small random perturbations by additive white-noise of a spatial discretization of a reaction–diffusion equation with a stable equilibrium and solutions that blow up in finite time. We prove that the perturbed system blows up with total probability and establish its order of magnitude and asymptotic distribution. For initial data in the domain of explosion we prove that the explosion time converges to the deterministic one while for initial data in the domain of attraction of the stable equilibrium we show that the system exhibits metastable behavior.     

Subcritical branching processes in a random environment without the Cramer condition

A subcritical branching process in random environment (BPRE) is considered whose associated random walk does not satisfy the Cramer condition. The asymptotics for the survival probability of the process is investigated, and a Yaglom type conditional limit theorem is proved for the number of particles up to moment $n$ given survival to this moment. Contrary to other types of subcritical BPRE, the limiting distribution is not discrete. We also show that the process survives for a long time owing to a single big jump of the associate random walk accompanied by a population explosion at the beginning of the process.     

A population explosion in an evolutionary game in spatial economics: Blow up radial solutions to the initial value problem for the replicator equation whose growth rate is determined by the continuous Dixit–Stiglitz–Krugman model in an urban setting

We consider a spatially continuous evolutionary game whose payoff is defined as the density of real wages that is determined by the continuous Dixit–Stiglitz–Krugman model in an urban setting. This evolutionary game is expressed by the initial value problem for the replicator equation whose growth rate contains an operator which acts on an unknown function that denotes the density of workers. We prove that this initial value problem has a unique global solution, and that if workers are distributed radially in space and concentrated in the neighborhood of a city at the initial time, then all workers will move toward the center of the city in such a way that the density of workers converges to the Dirac delta function in the sense of distribution. In the real world this result describes a population explosion caused by concentration of workers motivated by the disparity in real wages.     

Stochastic Gilpin-Ayala competition model with infinite delay

In this paper we study the stochastic Gilpin-Ayala competition model with an infinite delay. We verify that the environmental noise included in the model does not only provide a positive global solution (there is no explosion in a finite time), but this solution is also stochastically ultimately bounded. We obtain certain asymptotic results regarding a large time behavior.     

Distribution Function of the Blow up Time of the Solution of an Anticipating Random Fatigue Equation

In this paper, we study the distribution function of the time of explosion of a stochastic differential equation modeling the length of the dominant crack due to fatigue.The main novelty is that initial condition is regarded as an anticipating random variable and the stochastic integral is in the forward sense.Under suitable conditions, we use the substitution formula from Russo and Vallois to find the local solution of this equation.Then, we find the law of blow up time by proving some results on barrier crossing probabilities of Brownian bridge.     

The time evolution of the size distribution of droplets effects on the thermal explosion of polydisperse fuel spray

In this paper we investigate the problem of thermal explosion in a two-phase polydisperse combustible mixture (oxygen and fuel concentrations are takes into account). The current work presents a new, simplified model of the thermal explosion in a combustible gaseous mixture containing vaporizing fuel droplets of different radii (polydisperse). The polydispersity is modeled using a probability density function (PDF). The evolution of the size distribution of droplets due to the evaporation process is described by the kinetic equation for the PDF. An explicit expression of the critical condition for thermal explosion limit is derived analytically and represents a generalization of the critical parameter of the classical Semenov theory.     

A probabilistic model of thermal explosion in polydisperse fuel spray

This work is concerned with an analysis of polydisperse spray droplets distribution on the thermal explosion processes. In many engineering applications it is usual to relate to the practical polydisperse spray as a monodisperse spray. The Sauter Mean Diameter (SMD) and its variations are frequently used for this purpose [13]. The SMD and its modifications depend only on “integral” characterization of polydisperse sprays and can be the same for very different types of polydisperse spray distributions.The current work presents a new, simplified model of the thermal explosion in a combustible gaseous mixture containing vaporizing fuel droplets of different radii (polydisperse). The polydispersity is modeled using a probability density function (PDF) that corresponds to the initial distribution of fuel droplets size. This approximation of polydisperse spray is more accurate than the traditional ‘parcel’ approximation and permits an analytical treatment of the simplified model. Since the system of the governing equations represents a multi-scale problem, the method of invariant (integral) manifolds is applied.An explicit expression of the critical condition for thermal explosion limit is derived analytically. Numerical simulations demonstrate an essential dependence of these thermal explosion conditions on the PDF type and represent a natural generalization of the thermal explosion conditions of the classical Semenov theory.     

Damage analysis of arch concrete dams subjected to underwater explosion

The nonlinear behavior of arch concrete dams subjected to underwater explosions was investigated. Pressure waves were spherically modeled using a 3-D finite element method. The nonlinear behavior of concrete was modeled using the concrete damage plasticity model. In addition, absolute wave formulation was used to develop a fully-coupled model incorporating the propagation of the shock wave and water–dam interaction. Analysis of an arch concrete dam subject to underwater explosion was performed upon model verification. The dynamic response of the dam subject to the underwater explosion was studied for different sizes of explosions at different depths. The results showed that the closer the point to the explosive source, the sooner the impression and damage was caused by the blast. It was clear that the middle part of the dam facing the explosion was the first location to incur damage and then cracks extended to the downstream face and abutments. Finally, cracks extended to the bottom of the dam. The results of the dynamic analysis and dissipated energy by the loss of elements’ resistance indicated that the time slice of 150 ms was appropriate to analyze arch concrete dam structure subject to an underwater explosion. In addition, for a given amount of explosives, an increase in the depth of explosion corresponded to an increase in the maximum displacement of the upstream face of the dam. The results also confirmed that the damage distribution to the arch concrete dam subject to an underwater explosion depended upon the amount and depth of explosive materials.     

Well-posedness,global solutions and blowup phenomena for a nonlinearly dispersive wave equation

We establish the local well-posedness for a new nonlinearly dispersive wave equation which has solutions that exist for indefinite times as well as solutions that blowup infinite time. We also derive an explosion criterion for the equation, and we give a sharp estimate of the existence time for solutions with smooth initial data.     

Exponential Functionals of Brownian Motion and Explosion Times of a System of Semilinear SPDEs

We investigate lower and upper bounds for the blowup times of a system of semilinear SPDEs. Under certain conditions on the system parameters, we obtain explicit solutions of a related system of random PDEs, which allows us to use a formula due to Yor to obtain the distribution functions of several explosion times. We also give the Laplace transforms at independent exponential times of related exponential functionals of Brownian motion.     

基于分支定界算法的物流配送网络优化研究

为进一步提高物流配送网络的运行效率,以配送总里程最短为目标,建立了单配送中心的配送优化模型,提出了一种基于分枝定界法的混合求解策略。该策略能有效地避免智能启发式算法的不稳定性和传统优化算法的指数爆炸问题,对现代配送网络的建设具有一定的理论和实践意义。     

结构函数递推公式的证明及其在爆破网路可靠性计算中的应用

运用结构函数解决了常见爆破网路的可靠性计算的实际问题。进而提出并证明了结构函数的递推公式，实现了在计算机上计算常见网路的可靠性的目的。最后给出了表示网路可靠性的数量指标及其计算公式。     

Successive Canard Explosions in a Singularly Perturbed Spruce-Budworm Model with Holling-II Functional Response

By combining geometric singular perturbation theory (GSPT) with qualitative method, this paper analyzes the phenomenon of successive canard explosions in a singularly perturbed Spruce-Budworm model with Holling-II functional response. We select suitable parameters such that the critical curve is $S$-shaped, and the full model only admits a unique equilibrium. Then, under the variation of the breaking parameter, it is found that a canard explosion followed by an inverse canard explosion successively occurs in this model. That is, a relaxation oscillation arises via the first canard explosion, which persists for a large interval of parameter until it vanishes via the so-called inverse canard explosion. All these theoretical predictions are verified by numerical simulations.     

Metastability for small random perturbations of a PDE with blow-up

We study random perturbations of a reaction–diffusion equation with a unique stable equilibrium and solutions that blow-up in finite time. If the strength of the perturbation $\epsilon >0$ is small and the initial data is in the domain of attraction of the stable equilibrium, the system exhibits metastable behavior: its time averages remain stable around this equilibrium until an abrupt and unpredictable transition occurs which leads to explosion in a finite time (but exponentially large in ${\epsilon }^{?2}$). Moreover, for initial data in the domain of explosion we show that the explosion times converge to the one of the deterministic solution.     

Analysis of the dynamics of fuel spray using asymptotic methods: The method of integral invariant manifolds

The current research deals with the thermal explosion and ignition of a mixture of carbon and air. The size distribution of the carbon particles is taken to be continuous and is characterized by a probability density function. The chemical reaction term is presented in the Arrhenius form with variable pre-exponential factor. Transforming the new model to a dimensionless form enables us to rewrite the model in a singular perturbed system of ordinary differential equations. This form of the model enables us to apply the method of integral manifold (MIM). As a result of this method we can derive an explicit expression for the thermal explosion limit which depends on the initial probability density function. Comparing our numerical results to the analytical results, we observe that the effect of the thermal radiation is significant, especially at high temperatures, and cannot be ignored in the analysis of the phenomena of the explosion and ignition.     

Small-noise limit of the quasi-Gaussian log-normal HJM model

Quasi-Gaussian HJM models are a popular approach for modeling the dynamics of the yield curve. This is due to their low dimensional Markovian representation, which greatly simplifies their numerical implementation. We present a qualitative study of the solutions of the quasi-Gaussian log-normal HJM model. Using a small-noise deterministic limit we show that the short rate may explode to infinity in finite time. This implies the explosion of the Eurodollar futures prices in this model. We derive explicit explosion criteria under mild assumptions on the shape of the yield curve.     

Guaranteed approximation of Markov chains with applications to multiplexer engineering in ATM networks

Discrete Markov chains are applied widely for analysis and design of high speed ATM networks due to their essentially discrete nature. Unfortunately, their use is precluded for many important problems due to explosion of the state space cardinality. In this paper we propose a new method for approximation of a discrete Markov chain by a chain of considerably smaller dimension which is based on the duality theory of optimization. A novel feature of our approach is that it provides guaranteed upper and lower bounds for the performance indices defined on the steady state distribution of the original system. We apply our method to the problem of engineering multiplexers for ATM networks.     

Hill‐climbing to Pasch valleys

Exhaustive enumeration of Steiner Triple Systems is not feasible, due to the combinatorial explosion of instances. The next‐best hope is to quickly find a sample that is representative of isomorphism classes. Stinson's Hill‐Climbing algorithm [ 20 ] is widely used to produce random Steiner Triple Systems, and certainly finds a sample of systems quickly, but the sample is not uniformly distributed with respect to the isomorphism classes of STS with ν ≤ 19, and, in particular, we find that isomorphism classes with a large number of Pasch configurations are under‐represented. No analysis of the non‐uniformity of the distribution with respect to isomorphism classes or the intractability of obtaining a representative sample for ν > 19 is known. We also exhibit a modification to hill‐climbing that makes the sample if finds closer to the uniform distribution over isomorphism classes in return for a modest increase in running time. © 2007 Wiley Periodicals, Inc. J Combin Designs 15: 405–419, 2007     

Lyapunov exponent of a stochastic SIRS model

We consider a SIRS (susceptible–infected–removed–susceptible) model influenced by random perturbations. We prove that the solutions are positive for positive initial conditions and are global, that is, there is no finite explosion time. We present necessary and sufficient conditions for the almost sure asymptotic stability of the steady state of the stochastic system.