Numerical Optimal Harvesting for an Age-Dependent Prey-Predator System

In this article, we develop a numerical study of an optimal harvesting problem for age-dependent prey-predator system. Here, the rates of growth and decay as well as the interaction effect between species are assumed to be depending on age, time and space. Existence, uniqueness, and necessary conditions for the optimal control are assured in case of a small final time T. The discrete parabolic nonlinear dynamical systems are obtained by using a finite difference semi-implicit scheme. Then a numerical algorithm is developed to approximate the optimal harvesting effort and the optimal harvest. Results of the numerical tests are given.     

Numerical analysis and applications of Fokker-Planck equations for stochastic dynamical systems with multiplicative α-stable noises

In this paper, we study the nonlocal Fokker-Planck equations (FPEs) associated with Lévy-driven scalar stochastic dynamical systems. We first derive the Fokker-Planck equation for the case of multiplicative symmetric α-stable noises, by the adjoint operator method. Then we construct a finite difference scheme to simulate the nonlocal FPE on either bounded or infinite domain. It is shown that the semi-discrete scheme satisfies the discrete maximum principle and converges. Some experiments are conducted to validate the numerical method. Finally, we extend the results to the asymmetric case and present an application to the nonlinear filtering problem.     

The dynamical behavior of the discontinuous Galerkin method and related difference schemes

We study the dynamical behavior of the discontinuous Galerkin finite element method for initial value problems in ordinary differential equations. We make two different assumptions which guarantee that the continuous problem defines a dissipative dynamical system. We show that, under certain conditions, the discontinuous Galerkin approximation also defines a dissipative dynamical system and we study the approximation properties of the associated discrete dynamical system. We also study the behavior of difference schemes obtained by applying a quadrature formula to the integrals defining the discontinuous Galerkin approximation and construct two kinds of discrete finite element approximations that share the dissipativity properties of the original method.

     

Finite difference reaction diffusion equations with nonlinear boundary conditions

This article is concerned with numerical solutions of finite difference systems of reaction diffusion equations with nonlinear internal and boundary reaction functions. The nonlinear reaction functions are of general form and the finite difference systems are for both time-dependent and steady-state problems. For each problem a unified system of nonlinear equations is treated by the method of upper and lower solutions and its associated monotone iterations. This method leads to a monotone iterative scheme for the computation of numerical solutions as well as an existence-comparison theorem for the corresponding finite difference system. Special attention is given to the dynamical property of the time-dependent solution in relation to the steady-state solutions. Application is given to a heat-conduction problem where a nonlinear radiation boundary condition obeying the Boltzmann law of cooling is considered. This application demonstrates a bifurcation property of two steady-state solutions, and determines the dynamic behavior of the time-dependent solution. Numerical results for the heat-conduction problem, including a test problem with known analytical solution, are presented to illustrate the various theoretical conclusions. © 1995 John Wiley & Sons, Inc.     

Optimal control of stochastic differential equations with dynamical boundary conditions

In this paper we investigate the optimal control problem for a class of stochastic Cauchy evolution problems with nonstandard boundary dynamic and control. The model is composed by an infinite dimensional dynamical system coupled with a finite dimensional dynamics, which describes the boundary conditions of the internal system. In other terms, we are concerned with nonstandard boundary conditions, as the value at the boundary is governed by a different stochastic differential equation.     

LONG-TIME BEHAVIOR OF FINITE DIFFERENCE SOLUTIONS OF THREE-DIMENSIONAL NONLINEAR SCHRODINGER EQUATION WITH WEAKLY DAMPED

The three-dimensional nonlinear SchrSdinger equation with weakly damped that possesses a global attractor are considered. The dynamical properties of the discrete dynamical system which generate by a class of finite difference scheme are analysed. The existence of global attractor is proved for the discrete dynamical system.     

Dynamics in numerics: on two different finite difference schemes for ODEs

We compare two finite difference schemes for Kolmogorov type of ordinary differential equations: Euler's scheme (a derivative approximation scheme) and an integral approximation (IA) scheme, from the view point of dynamical systems. Among the topics we investigate are equilibria and their stability, periodic orbits and their stability, and topological chaos of these two resulting nonlinear discrete dynamical systems.     

On the expressiveness and decidability of o-minimal hybrid systems

This paper is driven by a general motto: bisimulate a hybrid system by a finite symbolic dynamical system. In the case of o-minimal hybrid systems, the continuous and discrete components can be decoupled, and hence, the problem reduces in building a finite symbolic dynamical system for the continuous dynamics of each location. We show that this can be done for a quite general class of hybrid systems defined on o-minimal structures. In particular, we recover the main result of a paper by G. Lafferriere, G.J. Pappas, and S. Sastry, on o-minimal hybrid systems. We also provide an analysis and extension of results on decidability and complexity of problems and constructions related to o-minimal hybrid systems.     

不确定非自治混沌陀螺仪在非线性输入下的有限时间稳定性

陀螺仪是一个非常有趣,又是永恒的非线性非自治动力系统课题,它可以显示出非常复杂的动力学行为,如混沌现象.在一个给定的有限时间内,研究非线性非自治陀螺仪鲁棒稳定性问题.假设陀螺仪系统受到模型不确定的外部扰动而摄动,系统参数并不知道,同时考虑了非线性输入的影响.为未知参数提出了适当的自适应律.以自适应律和有限时间控制理论为基础,提出非连续有限时间控制理论,来研究系统的有限时间稳定性.解析证明了闭循环系统的有限时间稳定性及其收敛性.若干数值仿真结果表明,该文的有限时间控制法是有效的,同时验证了该文的理论结果.     

LONG-TIME BEHAVIOR OF FINITE DIFFERENCE SOLUTIONS OF A NONLINEAR SCHRODINGER EQUATION WITH WEAKLY DAMPED

1. IntroductionThe nonlinear schr~r equation with weakly dampedwhere t = N, o > 0, together with appropriate boUndary and hatal condition, is ared inmany physical fields. The echtence of an attractor is one of the most boortant ~eristiCSfor a dissipative system. The long-tabs dynamics is completely determined by the attractorof the system. J.M. Ghidaglia[1] studied the lOng-the behavior of the nonlineaz Sequation (1.1) and proved the eAstence of a compact global attractor A in H'(n) which…     

Long-time convergence of numerical approximations for 2D GBBM equation

We study the long-time behavior of the finite difference solution to the generalized BBM equation in two space dimensions with dirichlet boundary conditions. The unique solvability of numerical solution is shown. It is proved that there exists a global attractor of the discrete dynamical system. Finally, we obtain the long-time stability and convergence of the difference scheme. Our results show that the difference scheme can effectively simulate the infinite dimensional dynamical systems. Numerical experiment results show that the theory is accurate and the schemes are efficient and reliable.     

Differential equations and solution of linear systems

Many iterative processes can be interpreted as discrete dynamical systems and, in certain cases, they correspond to a time discretization of differential systems. In this paper, we propose to derive iterative schemes for solving linear systems of equations by modeling the problem to solve as a stable state of a proper differential system; the solution of the original linear problem is then computed numerically by applying a time marching scheme. We discuss some aspects of this approach, which allows to recover some known methods but also to introduce new ones. We give convergence results and numerical illustrations. AMS subject classification 65F10, 65F35, 65L05, 65L12, 65L20, 65N06     

Long-time behavior of the difference solutions to the 2D GKS equation

We study the long-time behavior of the finite difference solution to the generalized Kuramoto-Sivashinsky equation in two space dimensions with periodic boundary conditions. The unique solvability of numerical solution is shown. It is proved that there exists a global attractor of the discrete dynamical system and the upper semicontinuity d(A_h,τ,A)→0. Finally, we obtain the long-time stability and convergence of the difference scheme. Our results show that the difference scheme can effectively simulate the infinite dimensional dynamical systems.     

Upper Semicontinuity and Kolmogorov ε-Entropy of Global Attractor for κ-Dimensional Lattice Dynamical System Corresponding to Klein-Gordon-SchrSdinger Equation

In this paper, we establish the existence of a global attractor for a coupled κ-dimensional lattice dynamical system governed by a discrete version of the Klein-Gordon-SchrSdinger Equation. An estimate of the upper bound of the Kohnogorov ε-entropy of the global attractor is made by a method of element decomposition and the covering property of a polyhedron by balls of radii ε in a finite dimensional space. Finally, a scheme to approximate the global attractor by the global attractors of finite-dimensional ordinary differential systems is presented .     

Stability properties of a nonstandard finite difference scheme for a hantavirus epidemic model

This paper studies the stability properties of a nonstandard finite difference (NSFD) scheme used to simulate the dynamics of a mouse population model in hantavirus epidemics. It is shown that this difference scheme and the underlying system of differential equations have the same dynamics. The proof uses the fact that the total population obeys the logistic equation, as well as techniques from calculus, graphical analysis, and dynamical systems.     

一类三维抛物型方程有限差分逼近的长时间性态

1引言 抛物型方程是一类十分重要的方程,它出现在很多数学物理问题中,对这类方程的研究已有大量工作,如[10-12]等.随着无穷维动力系统研究的深入,人们越来越关心系统的长时间性态,而追踪系统长时间性态很大程度上依赖数值计算.     

Long-Time Behavior of Finite Difference Solutions of a Nonlinear Schrödinger Equation with Weakly Damped

A weakly damped Schrödinger equation possessing a global attractor are considered. The dynamical properties of a class of finite difference scheme are analysed. The existence of global attractor is proved for the discrete system. The stability of the difference scheme and the error estimate of the difference solution are obtained in the autonomous system case. Finally, long-time stability and convergence of the class of finite difference scheme also are analysed in the nonautonomous system case.     

Gene expression from 2-adic dynamical systems

We perform geometrization of genetics by representing genetic information by points of the 4-adic information space. By a well-known theorem of number theory this space can also be represented as the 2-adic space. The process of DNA reproduction is described by the action of a 4-adic (or equivalently 2-adic) dynamical system. As we know, the genes contain information for production of proteins. The genetic code is a degenerate map of codons to proteins. We model this map as the functioning of a polynomial dynamical system. The purely mathematical problem under consideration is to find a dynamical system reproducing the degenerate structure of the genetic code. We present one of possible solutions of this problem.     

Applying a Power Penalty Method to Numerically Pricing American Bond Options

In this paper, we aim to develop a numerical scheme to price American options on a zero-coupon bond based on a power penalty approach. This pricing problem is formulated as a variational inequality problem (VI) or a complementarity problem (CP). We apply a fitted finite volume discretization in space along with an implicit scheme in time, to the variational inequality problem, and obtain a discretized linear complementarity problem (LCP). We then develop a power penalty approach to solve the LCP by solving a system of nonlinear equations. The unique solvability and convergence of the penalized problem are established. Finally, we carry out numerical experiments to examine the convergence of the power penalty method and to testify the efficiency and effectiveness of our numerical scheme.     

带有多项式非线性项的高维反应扩散方程有限差分格式的长时间行为

本文考虑了一类带有多项式非线性项的高维反应扩散方程．建立了一个全离散的有限差分格式,并证明了差分解的存在唯一性．分析了由差分格式生成的离散系统的动力性质,在对差分解先验估计的基础上得到了离散动力系统的整体吸引子的存在性．最后证明了差分格式的长时间稳定性和收敛性．