A Shadowing Theorem for ordinary differential equations

A new notion of shadowing of a pseudo orbit, an approximate solution, of an autonomous system of ordinary differential equations by an associated nearby true orbit is introduced. Then a general theorem which guarantees the existence of shadowing of pseudo orbits in compact hyperbolic sets is proved.Supported in part by the Air Force.Supported in part by NSF grant DMS 9201951.     

Long periodic shadowing

A general theorem for establishing the existence of a true periodic orbit near a numerically computed pseudoperiodic orbit of an autonomous system of ordinary differential equations is presented. For practical applications, a Newton method is devised to compute appropriate pseudoperiodic orbits. Then numerical considerations for checking the hypotheses of the theorem in terms of quantities which can be computed directly from the pseudoperiodic orbit and the vector field are addressed. Finally, a numerical method for estimating the Lyapunov exponents of the true periodic orbit is given. The theory and computations are designed to be applicable for unstable periodic orbits with long periods. The existence of several such periodic orbits of the Lorenz equations is exhibited. This revised version was published online in June 2006 with corrections to the Cover Date.     

Periodic solutions for ordinary differential equations with sublinear impulsive effects

The continuation method of topological degree is used to investigate the existence of periodic solutions for ordinary differential equations with sublinear impulsive effects. The applications of the abstract approach include the generalizations of some classical nonresonance theorem for impulsive equations, for instance, the existence theorem for asymptotically positively homogeneous differential systems and the existence theorem for second order equations with Landesman-Lazer conditions.     

Transversal connecting orbits from shadowing

A rigorous numerical method for establishing the existence of a transversal connecting orbit from one hyperbolic periodic orbit to another of a differential equation in is presented. As the first component of this method, a general shadowing theorem that guarantees the existence of such a connecting orbit near a suitable pseudo connection orbit given the invertibility of a certain operator is proved. The second component consists of a refinement procedure for numerically computing a pseudo connecting orbit between two pseudo periodic orbits with sufficiently small local errors so as to satisfy the hypothesis of the theorem. The third component consists of a numerical procedure to verify the invertibility of the operator and obtain a rigorous upper bound for the norm of its inverse. Using this method, existence of chaos is demonstrated on examples with transversal homoclinic orbits, and with cycles of transversal heteroclinic orbits.     

Monotone positive solutions of second-order nonlinear differential equations

We obtain an existence theorem for monotone positive solutions of nonlinear second-order ordinary differential equations by using the Schauder–Tikhonov fixed point theorem. The result can also be applied to prove the existence of positive solutions of certain semilinear elliptic equations in $\text{R}{}^{\mathrm{n}}\text{(n⩾3)}$.     

Solvability of a system of equations for the Fourier–Chebyshev coefficients when solving ordinary differential equations by the Chebyshev series method

A solvability theorem for a system of equations with respect to approximate values of Fourier–Chebyshev coefficients is formulated. This theorem is a theoretical justification for numerical solution of ordinary differential equations using Chebyshev series.     

Justification of an Approach to Application of Orthogonal Expansions for Approximate Integration of Canonical Systems of Second Order Ordinary Differential Equations

A solvability theorem for a nonlinear system of equations with respect to approximate values of Fourier—Cliebysliev coefficients is proved. This theorem is a theoretical substantiation for the numerical solution of second order ordinary differential equations using Chebyshev series and Markov quadrature formulas.     

Immersed boundary method: The existence of approximate solution of the two-dimensional heat equation

This paper deals with the heat equation in which the source term involves a Dirac function and describes the elastic reaction of the immersed boundary. We analyze the existence of the approximate solution in two-dimensional case with Dirac function approximated by differentiable function. We obtain the result via finite element method, the Banach Fixed-Point Theorem and a theorem in nonlinear ordinary differential equations in abstract space.     

On uniform attractors of explicit approximations

We prove a theorem stating that the uniform attractors of a family of semiprocesses that do not necessarily have a common time semigroup depend on the parameter uppersemicontinuously. We consider an explicit finite-difference scheme for a nonautonomous system of ordinary differential equations and an explicit spectral-difference scheme for the vorticity equation with time-dependent bounded right-hand side on a sphere. We obtain theorems on the existence of uniform attractors of numerical schemes and their closeness to true attractors of the original differential problems.     

Perturbation method for nonlocal impulsive evolution equations

This paper deals with the existence of mild solutions for a class of semilinear nonlocal impulsive evolution equations in ordered Banach spaces. The existence and uniqueness theorem of mild solution for the associated linear nonlocal impulsive evolution equation is established. With the aid of the theorem, the existence of mild solutions for nonlinear nonlocal impulsive evolution equation is obtained by using perturbation method and monotone iterative technique. The theorems proved in this paper improve and extend some related results in ordinary differential equations and partial differential equations. Moreover, we present two examples to illustrate the feasibility of our abstract results.     

中立型泛函微分方程的周期解

对于中立型泛函微分方程,证明了解的毕竟有界性蕴含周期解的存在性,把常微分方程中著名的Yoshizawa周期解存在定理推广到中立型泛函微分方程,然后利用所得结果给出一类产生于电力系统的中立型时滞泛函微分方程周期解存在惟一与吸引的条件。     

Quadratic backward stochastic differential equations driven by G-Brownian motion: Discrete solutions and approximation

In this paper, we consider backward stochastic differential equations driven by $G$-Brownian motion (GBSDEs) under quadratic assumptions on coefficients. We prove the existence and uniqueness of solution for such equations. On the one hand, a priori estimates are obtained by applying the Girsanov type theorem in the $G$-framework, from which we deduce the uniqueness. On the other hand, to prove the existence of solutions, we first construct solutions for discrete GBSDEs by solving corresponding fully nonlinear PDEs, and then approximate solutions for general quadratic GBSDEs in Banach spaces.     

一个非线性常微分方程周期解的存在定理

该文通过应用Miranda定理,证明了在R~n空间某有界区域的边界上一些不等式成立时,非线性微分方程x=f(x,t)在此区域内存在周期解的定理。     

Infinite Systems of Quasilinear Differential Difference Inequalities and Applications

The article deals with the initial boundary value problem for an infinite system of first order quasilinear functional differential equations. A comparison result concerning infinite systems of differential difference inequalities is proved. A function satisfying such inequalities is estimated by a solution of a suitable Cauchy problem for an ordinary functional differential system. The comparison result is used in an existence theorem and in the investigation of the stability of the numerical method of lines for the original problem. A theorem on the error estimate of the method is given. The infinite system of first order functional differential equations contains, as particular cases, equations with a deviated argument and integral differential equations of the Volterra type.     

一类多时滞微分方程初值问题解的存在唯一性

研究了一类多时滞微分方程初值问题解的存在唯一性,用Picard方法证明了这类初值问题解的存在唯一性结论,它是常微分方程基本理论中著名的Picard存在唯一性定理的推广.     

Almost automorphic solutions for abstract functional differential equations

For abstract linear functional differential equations with an almost automorphic forcing term, we establish a result on the existence of almost automorphic solutions, which extends the classical theorem due to Massera on the existence of periodic solutions for linear periodic ordinary differential equations.     

Solvability of a coupled system of parabolic and ordinary differential equations

A model of coupled parabolic and ordinary differential equations for a heterogeneous catalytic reaction is considered and the existence and uniqueness theorem of the classic solution is proved.     

The bifurcation of homoclinic orbits from two heteroclinic orbits-a topological approach

Conditions are found for a homoclinic orbit to bifurcate from a heteroclinic loop for autonomous ordinary differential equations. The results axe applied to prove the existence of traveling wave solutions of the FitzHugh-Nagumo equations and a system of reaction diffusion equations which arise as a model for a two step combustion process.     

Existenzsätze mit der linienmethode für parabolische probleme und periodische lösungen von ut=f(t,x,u,ux,uxx)

The (longitudinal) method of lines transforms a parabolic equation into a first order system of ordinary differential equations by discretization of the spatial variable. It is shown how to obtain existence theorems for nonlinear parabolic equations from those for ordinary differential equations under general growth conditions and weak regularity assumptions. The method is demonstrated in proving a new existence theorem for periodic solutions to u_t=f(t,x,u,u_x,u_xx) with boundary conditions of Dirichlet type.     

Anti-periodic solutions for Rayleigh-type equations via the reproducing kernel Hilbert space method

In this paper, reproducing kernel theorem is employed to solve anti-periodic solutions for Rayleigh-type equations. A simple algorithm is given to obtain the approximate solutions of the equations. By comparing the approximate solution with the exact analytical solution, we find that the simple algorithm is of good accuracy and it can be also applied to some ordinary or partial differential equations with initial-boundary value conditions and nonlocal boundary value conditions.