Proof and generalization of Kaplan-Yorke’s conjecture under the condition f ' (0) > 0 on periodic solution of differential delay equations

Using the theory of existence of periodic solutions of Hamiltonian systems, it is shown that many periodic solutions of differential delay equations can be yielded from many families of periodic solutions of the coupled generalized Hamiltonian systems. Some sufficient conditions on the existence of periodic solutions of differential delay equations are obtained. As a corollary of our results, the conjecture of Kaplan-Yorke on the search for periodic solutions for certain special classes of scalar differential delay equations is shown to be true when . Project supported by the National Natural Science Foundation of China (Grant No. 19731003) and Science Foundation of Yunnan Province.     

Global stability of the Armstrong-Frederick model with periodic biaxial inputs

The paper is concerned with the study of plasticity models described by differential equations with stop and play operators. We suggest sufficient conditions for the global stability of a unique periodic solution for the scalar models and for the vector models with biaxial inputs of a particular form, namely the sum of a uniaxial function and a constant term. For another class of simple biaxial inputs, we present an example of the existence of unstable periodic solutions. The paper was written during the research stay of D. Rachinskii at the Technical University Munich supported by the research fellowship from the Alexander von Humboldt Foundation. His work was partially supported by the Russian Science Support Foundation, Russian Foundation for Basic Research (Grant No. 01-01-00146, 03-01-00258), and the Grants of the President of Russia (Grant No. MD-87.2003.01, NS-1532.2003.1). The support is gratefully acknowledged.     

Blow-up Results for Some Nonlinear Delay Differential Equations

In this work we study the blow up phenomena for some scalar delay differential equations. In particular, we make connection with the blow up of ordinary differential equations that are related to the delay differential equations. The first author is supported by a Grant from TWAS under contract No: 03-030 RG/MATHS/AF/AC. The second author is supported by a grant from the Lebanese National Council for Scientific Research.     

A note on a generalization of Diliberto’s Theorem for certain differential equations of higher dimension

In the theory of autonomous perturbations of periodic solutions of ordinary differential equations the method of the Poincaré mapping has been widely used. For the analysis of properties of this mapping in the case of two-dimensional systems, a result first obtained probably by Diliberto in 1950 is sometimes used. In the paper, this result is (partially) extended to a certain class of autonomous ordinary differential equations of higher dimension.This research was supported by Grant No. 201/99/0295 of the Grant Agency of the Czech Republic.This revised version was published online in April 2005 with a corrected missing date string.     

Periodic solutions of abstract neutral functional differential equations with infinite delay

We establish the existence of mild solutions and periodic mild solutions for a class of abstract first-order non-autonomous neutral functional differential equations with infinite delay in a Banach space. This research was supported by FONDECYT-CONICYT, Grant 1050314.     

Normal forms for periodic orbits of real vector fields

We consider normal forms of real vector fields near periodic orbits and provide sufficient conditions for their smooth linearization. In addition, the main results also assert the existence of vertical local foliations, whose leaves are all transversal to the periodic orbit.     

Maximum principle of optimal control for functional differential systems

We show that the maximum principle holds for optimal periodic control problems governed by functional differential equations under a Lipschitz condition on the value functional. Generalizations to other boundary conditions are also considered.This research was partially supported by NSF Grant No. DMS-84-01719.The first author was partially supported by the Science Fund of the Chinese Academy of Sciences, Beijing, China.     

On numerical approximation using differential equations with piecewise-constant arguments

In this paper we give a brief overview of the application of delay differential equations with piecewise constant arguments (EPCAs) for obtaining numerical approximation of delay differential equations, and we show that this method can be used for numerical approximation in a class of age-dependent population models. We also formulate an open problem for stability and oscillation of a class of linear delay equations with continuous and piecewise constant arguments. This research was partially supported by Hungarian NFSR Grant No. T046929.     

On Differential Equations with Retarded Argument

We study the limit properties of solutions to one class of systems of differential equations as the number of equations and some parameters tend to infinity. We establish a close connection between solutions to these systems and differential equations with retarded argument. Our convergence theorems provide a new method for approximation of solutions to nonlinear differential equations with retarded argument.Original Russian Text Copyright © 2005 Demidenko G. V. and Likhoshvai V. A.The authors were supported by the Russian Foundation for Basic Research (Grants 03-01-00328, 03-04-48506, 03-04-48829, 03-07-96833, 04-01-00458, 05-04-49068, and 05-07-90274), the Integration Grant of the Siberian Branch of the Russian Academy of Sciences (No. 119 and 148), the Program Development of the Scientific Potential of Higher School of the Ministry for Education of the Russian Federation (Grant 8273), and NSF:FIBR (Grant EF-0330786).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 538–552, May–June, 2005.     

A note on solutions of orientor fields for periodic systems

It is shown that the orientor field problem can be solved for control problems governed by ordinary differential equations where periodic controls give periodic trajectories.This research was partially supported by NSF Grant No. GP-27973.     

Positivity and stability for some partial functional differential equations

The aim of this work is to study the stability for some linear partial functional differential equations. We assume that the linear part is non-densely defined and satisfies the Hille-Yosida condition. Using the positiveness, we give nessecary and sufficient conditions independently of the delay to ensure the uniform exponential stability of the solution semigroup. An application is given for a reaction diffusion equation with several delays. RID="h1" ID="h1"This work is supported by the Moroccan Grant PARS MI 36 and TWAS Grant under contract: No. 00-412 RG/MATHS/AF/AC.     

Homogeneous linear differential equations with only periodic solutions

We determine all third order homogeneous linear differential equations with periodic coefficients and only periodic solutions. The method extends tonth order equations. As an application, we show that the Laguerre-Forsyth canonical form cannot be used for global investigations i projective differential geometry. Research supported by NSF Grant GP-8176. To the Memory of Eri Jabotinsky     

Numerical Computation of Connecting Orbits in Delay Differential Equations

We discuss the numerical computation of homoclinic and heteroclinic orbits in delay differential equations. Such connecting orbits are approximated using projection boundary conditions, which involve the stable and unstable manifolds of a steady state solution. The stable manifold of a steady state solution of a delay differential equation (DDE) is infinite-dimensional, a problem which we circumvent by reformulating the end conditions using a special bilinear form. The resulting boundary value problem is solved using a collocation method. We demonstrate results, showing homoclinic orbits in a model for neural activity and travelling wave solutions to the delayed Hodgkin–Huxley equation. Our numerical tests indicate convergence behaviour that corresponds to known theoretical results for ODEs and periodic boundary value problems for DDEs.     

Almost periodic solutions for undamped nonhomogeneous delay-differential equations

We first establish a result giving conditions that certain undamped delay differential equations with almost periodic time dependence have unique almost periodic solutions. Using this result we obtain conditions that a second order scalar nonlinear delay differential equation with almost periodic forcing will have a unique almost periodic solution having saddle-type stability properties. These results use the method of averaging.

     

Local structure-preserving algorithms for partial differential equations

In this paper, we discuss the concept of local structure-preserving algorithms (SPAs) for partial differential equations, which are the natural generalization of the corresponding global SPAs. Local SPAs for the problems with proper boundary conditions are global SPAs, but the inverse is not necessarily valid. The concept of the local SPAs can explain the difference between different SPAs and provide a basic theory for analyzing and constructing high performance SPAs. Furthermore, it enlarges the applicable scopes of SPAs. We also discuss the application and the construction of local SPAs and derive several new SPAs for the nonlinear Klein-Gordon equation. This work was supported by the National Basic Research Program (Grant No. 2005CB321703). The first author was supported by the National Natural Science Foundation of China (Grant Nos. 40405019, 10471067) and the Major Research Projects of Jiangsu Province (Grant No. BK2006725); the second author was supported by the National Natural Science Foundation of China (Innovation Group) (Grant No. 40221503) and the third author was supported by the National Natural Science Foundation of China (Grant No. 10471145)     

Existence and stability of large scale nonlinear oscillations in suspension bridges

A nonlinear model of a suspension bridge is considered in which large-scale, stable oscillatory motions can be produced by constant loading and a small-scale, external oscillatory force. Loud's implicit-function theoretic method for determining existence and stability of periodic solutions or nonlinear differential equations is extended to a case of a non-differentiable nonlinearity.Author partially supported by NSF under Grant DMS 8318204 and AFOSR Grant 85-0330.Author partially supported by NSF under Grant DMS 9519882.Author partially supported by NSF under Grant DMS 8519776.     

A generalization of a minimax theorem of Fan via a theorem of the alternative

A minimization problem for a functional on a convex subsetC of a normed linear space is considered. Under certain hypotheses, optimality in a certain subset ofC implies the validity of first-order necessary optimality conditions for the problem inC. The result is applied to a problem in optimal periodic control of neutral functional differential equations.This work was partially supported by a grant from Deutsche Forschungsgemeinschaft and by AFOSR under Grant No. AFOSR-84-0398.     

Optimization of functional differential systems

A method of determining optimal control for functional differential systems is presented. This is a generalization of Krasovskii's results on the optimal control of time delay systems, which include several of the results published recently. A technique for the numerical solution of the resulting Riccati equations is developed, and an example is worked to illustrate the presented results.The research reported herein was partly supported by JSEP Contract No. F44620-71-C-0091 and AF Grant No. AFOSR-72-2371.     

Asymptotic behavior of a system of interacting nuclear-space-valued stochastic differential equations driven by Poisson random measures

In this paper we study a system of interacting stochastic differential equations taking values in duals of nuclear spaces driven by Poisson random measures. We also consider the McKean-Vlasov equation associated with the system. We show that under suitable conditions the system has a unique solution and the sequence of its empirical distributions converges to the solution of the McKean-Vlasov equation when the size of the system tends to infinity. The results are applied to the voltage potentials of a large system of neurons and the limiting distribution of the empirical measure is obtained.This research was supported by the National Science Foundation, the Air Force Office of Scientific Research under Grant No. F49620-92-J-0154, and the Army Research Office under Grant No DAAL03-92-G-0008.     

On Doubly Degenerate Quasilinear Parabolic Equations of Higher Order

We deal with the existence of periodic solutions for doubly degenerate quasilinear parabolic equations of higher order, which can degenerate, on a part of the boundary, on a segment in the interior of the domain and in time.