Heteroclinic Solutions for a System of Strongly Coupled ODEs

We use the implicit function theorem to prove an existence of a heteroclinic orbit to a system of two non-linear second-order ODEs. The perturbation is carried out around infinite value of a ‘coupling parameter’. The form of the system which is considered in this paper is related to the system defining travelling wave solutions in a two temperature model of the laser sustained plasma.     

A Singular Perturbation Analysis for a Reaction-Diffusion System Describing Pattern Formation

A reaction-diffusion system describing a simple activator-inhibitor reaction is investigated in the limiting case of “very large” diffusion rate of the inhibitor. Using singular perturbation techniques, an inner and an outer expansion are derived. The latter, describing the large-time behavior of the system, is governed by equations that, in the first orders of approximation, possess asymptotically stable space-inhomogeneous equilibria (patterns), and whose set of equilibria is globally attractive.     

A system of ODEs with a hysteresis effect

The paper deals with a class of ordinary differential systems which contains a differential inclusion describing input–output relations of hysteresis type. Existence and uniqueness of local and global solutions of the systems under consideration are proved.     

A new mesh selection strategy for ODEs

In this paper a new mesh selection strategy, based on the conditioning properties of continuous problems, is presented. It turns out to be particularly efficient when approximating solutions of BVPs. The numerical methods used to test the reliability of the strategy are symmetric Linear Multistep Formulae (LMF) used as Boundary Value Methods (BVMs) since they provide a wide choice of methods of arbitrary high order and have similar stability properties to each other. In particular, we shall consider a subclass of such methods, called Top Order Methods (TOMs) (Amodio, 1996; Brugnano and Trigiante, 1995, 1996), to carry out the numerical results on some singular perturbation test problems.     

A new analytical method for system of ODEs

In this article, a new homotopy perturbation method (NHPM) is introduced to obtain exact solutions of system of ODEs. Theoretical considerations are discussed. Comparison of the results of applying NHPM with those of homotopy perturbation method (HPM) and Adomia's decomposition method (ADM) leads to significant consequences. Three examples, including stiff system of linear and nonlinear ODEs are given to demonstrate the efficiency of the new method. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

关于多级一阶线性常系数微分方程组的一点注记

有O初始条件的多级一阶线性常系数微分方程组,可以用拉氏变换求解,并且解还可以用Green矩阵来表达.早已为人所知.但关于Green矩阵的一个在实际中有用的表达形式,因为推导上有毛病,未被普遍接受.本文对此毛病作了点注记.予以克服.     

常微分方程解的延伸定理的特殊形式

在本科数学专业的常微分方程课程中,解的延伸定理是一个重要的结论.在具体问题中,经常需要研究解在最大存在区间端点的极限.在特殊情形下,通过连续可导函数的特殊性,改进解的延伸定理.得到解在存在区间边界的极限性质.     

Analysis of a multirate theta-method for stiff ODEs

This paper contains a study of a simple multirate scheme, consisting of the θ-method with one level of temporal local refinement. Issues of interest are local accuracy, propagation of interpolation errors and stability. The theoretical results are illustrated by numerical experiments, including results for more levels of refinement with automatic partitioning.     

Global Controllability for Quasilinear Nonnegative Definite System of ODEs and SDEs

Journal of Optimization Theory and Applications - We consider exact and averaged control problem for a system of quasilinear ODEs and SDEs with a nonnegative definite symmetric matrix of the...     

Design of software for ODEs

We discuss the design of software that is easy to use for simple problems, but still capable of solving complicated problems. Nowadays a design should accommodate related, but fundamentally different, tasks such as solving DAEs and DDEs in addition to IVPs and BVPs for ODEs. Major issues are illustrated with the MATLAB ODE Suite and selected solvers written in Fortran 90 and Maple.     

Period two implies chaos for a class of ODEs

We extend a result of J. Andres and K. Pastor concerning scalar time-periodic first order ordinary differential equations without uniqueness, by proving that the existence of just one subharmonic implies the existence of large sets of subharmonics of all given orders. Since these periodic solutions must coexist with complicated dynamics, we might paraphrase T. Y. Li and J. A. Yorke by loosely saying that in this setting even period two implies chaos. Similar results are obtained for a class of differential inclusions.

     

Construction of a multirate RODAS method for stiff ODEs

Multirate time stepping is a numerical technique for efficiently solving large-scale ordinary differential equations (ODEs) with widely different time scales localized over the components. This technique enables one to use large time steps for slowly varying components, and small steps for rapidly varying ones. Multirate methods found in the literature are normally of low order, one or two. Focusing on stiff ODEs, in this paper we discuss the construction of a multirate method based on the fourth-order RODAS method. Special attention is paid to the treatment of the refinement interfaces with regard to the choice of the interpolant and the occurrence of order reduction. For stiff, linear systems containing a stiff source term, we propose modifications for the treatment of the source term which overcome order reduction originating from such terms and which we can implement in our multirate method.     

Projection with a Minimal System of Inequalities

Projection of a polyhedron involves the use of a cone whose extreme rays induce the inequalities defining the projection. These inequalities need not be facet defining. We introduce a transformation that produces a cone whose extreme rays induce facets of the projection.     

N-dark Soliton Solutions for the Multi-component Maccari System

By virtue of the KP hierarchy reduction method, the N-dark soli- ton solutions of the multi-component Maccari system are constructed. Taking the coupled Maccari system for instance, the N-dark soliton solutions are fur- ther obtained in terms of determinants. In addition, in contrast with bright- bright soliton collisions, the dynamical analysis shows that the collisions of dark-dark solitons are elastic and there is no energy exchange among soli- tons in different components. What’s more, we also investigate the dark-dark soliton bound states including stationary and moving ones.     

Soliton Solutions of a Generalization of the Coupled Volterra System

Computational Mathematics and Mathematical Physics - The possibility of finding soliton solutions of a nonintegrable generalization of the coupled Volterra system is studied. This generalization is...     

A Multiresolution Method for Numerical Reduction and Homogenization of Nonlinear ODEs

The multiresolution analysis (MRA) strategy for the reduction of a nonlinear differential equation is a procedure for constructing an equation directly for the coarse scale component of the solution. The MRA homogenization process is a method for building a simpler equation whose solution has the same coarse behavior as the solution to a more complex equation. We present two multiresolution reduction methods for nonlinear differential equations: a numerical procedure and an analytic method. We also discuss one possible appproach to the homogenization method.     

一阶系统非线性边值问题的奇摄动*

本文应用比较定理研究了一类非线性边界条件的向量非线性奇摄动问题εx='f(t,x,y,e)εy'=g(t,x,y,ε)x(0)=A(ξ₁,ξ₂,x(1)-x(0),y(1)-y(0),ε)y(0)=B(ξ₁,ξ,x(1)-x(0),y(1)-y(0),ε)这里ξ₁,ξ₂为ε的函数。0<ε<<1,在适当的条件下,作出了任意次精度的渐近展式。并得出余项估计。     

Normal Form of a Hamiltonian System with a Periodic Perturbation

Computational Mathematics and Mathematical Physics - A perturbed Hamiltonian system with a time-independent unperturbed part and a time-periodic perturbation is considered near a stationary...     

Period two implies all periods for a class of ODEs: A multivalued map approach

We present an elementary proof that, for a multivalued map with nonempty connected values and monotone margins, the existence of a periodic orbit of any order implies the existence of periodic orbits of all orders. This generalizes a very recent result of this type in terms of scalar ordinary differential equations without uniqueness, due to F. Obersnel and P. Omari, obtained by means of lower and upper solutions techniques.