Existence of dichotomies and invariant splittings for linear differential systems,II

This paper is primarily concerned with linear time-varying ordinary differential equations. Sufficient conditions are given for the existence of an exponential dichotomy or equivalently an invariant splitting. The conditions are more general than those given in Part I of this paper and include the case in which the coefficients lie in a base space which is chain-recurrent under the translation flow and also the case in which compatible splittings are known to exist over invariant subsets of the base space. When the compatibility fails, the flow in the base space is shown to exhibit a gradient-like structure with attractors and repellers. Sufficient conditions are given guaranteeing the existence of bounded solutions of a linear system. The problem is treated in the unified setting of a skew-product dynamical system and the results apply to discrete systems including those generated by diffeomorphisms of manifolds. Sufficient conditions are given for a diffeomorphism to be an Anosov diffeomorphism.     

Existence of dichotomies and invariant splittings for linear differential systems,III

This paper is primarily concerned with linear time-varying ordinary differential equations. Sufficient conditions are given for the existence of a “trichotomy,” i.e., a continuous decomposition of Rⁿ into stable, unstable and neutral subspaces. For constant coefficients it reduces to the usual (Jordan) decomposition of Rⁿ into subspaces corresponding to eigenvalues with negative, positive, and zero real parts, respectively, but only in the case in which the eigenvalues with zero real parts occur with simple elementary divisors. The conditions are related to those used by Favard in his study of almost periodic equations. The problem is treated in the unified setting of a skew-product dynamical system and the results apply to discrete systems including those generated by diffeomorphisms of manifolds. In the continuous case, sufficient conditions are given for a flow on a compact manifold to be an Anosov flow.     

A spectral theory for linear differential systems

This paper is concerned with continuous and discrete linear skew-product dynamical systems including those generated by linear time-varying ordinary differential equations. The concept of spectrum is introduced for a linear skew-product dynamical system. In the case of a system of ordinary differential equations with constant coefficients the spectrum reduces to the real parts of the eigenvalues. In the general case continuous spectrum can occur and under certain conditions it consists of finitely many compact intervals of the real line, their number not exceeding the dimension of the system. A spectral decomposition theorem is proved which says that a certain naturally defined vector bundle is the sum of invariant subbundles, each one associated with a spectral subinterval. This partially generalizes the Jordan decomposition in the case of constant coefficients. A perturbation theorem is proved which says that nearby systems have spectra which are close. Almost periodic systems are given special attention.     

Integral expressions of Lyapunov exponents for autonomous ordinary differential systems

In the paper, the author addresses the Lyapunov characteristic spectrum of an ergodic autonomous ordinary differential system on a complete riemannian manifold of finite dimension such as the d-dimensional euclidean space ℝ ^d , not necessarily compact, by Liaowise spectral theorems that give integral expressions of Lyapunov exponents. In the context of smooth linear skew-product flows with Polish driving systems, the results are still valid. This paper seems to be an interesting contribution to the stability theory of ordinary differential systems with non-compact phase spaces. This work was supported by the National Natural Science Foundation of China (Grant No. 10671088) and the Major State Basic Research Development Program of China (Grant No. 2006CB805903)     

Order-Preserving Skew-Product Flows and Nonautonomous Parabolic Systems

The paper deals with order-preserving (or monotone) skew-product flows in Banach spaces. Within a general framework, we study long-time behavior of their trajectories. We introduce the concepts of equilibria, sub- and super-equilibria and we give simple conditions that guarantee their existence. Attention is mainly paid to order-preserving skew-product flows which have an additional concavity property called sublinearity, frequently encountered in applications. For these flows we prove the uniqueness and stability of equilibria and also a limit set trichotomy, stating that either (i) all orbits are unbounded, or (ii) all orbits are bounded but their closure reaches out to the boundary of the part, or (iii) there exists a unique, globally attracting equilibrium. Our main examples are quasilinear systems of parabolic equations with almost-periodic coefficients.     

Unified boundedness,periodicity, and stability in ordinary and functional differential equations

Summary We discuss a unified theory of periodicity of dissipative ordinary and functional differential equations in terms of uniform boundedness. Sufficient conditions for the uniform boundedness are given by means of Liapunov functionals having a weighted norm as an upper bound. The theory is developed for ordinary differential equations, equations with bounded delay, and equations with infinite delay.On leave from Anhui University, Hefei, Anhui, People's Republic of China     

Systems of matrix linear differential equations of first order

Sufficient conditions for the solvability in quadratures of systems of matrix linear ordinary differential equations of first order with one-sided multiplication by variable matrix coefficients are given in this paper. These conditions are stated in terms of the theory of Lie algebras. We consider matrix equations of higher orders that are equivalent to such systems. An illustrative example is considered. Translated fromMatematicheskie Zametki, Vol. 66, No. 1, pp. 63–75, July, 1999.     

Tight estimates for general averaging applied to almost-periodic differential equations

Estimates are presented for the averaging method of ordinary differential equations. Previous results are improved by relaxing the conditions under which they hold, and by providing tight bounds for the estimations for almost-periodic differential equations and quasi-periodic differential equations.     

The spectrum of an invariant submanifold

This paper is concerned with vector fields on smooth compact manifolds. The exponential growth of solutions of the linearized equations is described by the already well-known Spectral Theorem applied to the induced linear flow on the tangent bundle. The spectrum of the tangent bundle flow is compared to the two secondary spectra obtained by first taking the spectrum of the bundle of tangent spaces to an invariant submanifold and second, the spectrum of an induced flow on an arbitrary complementary bundle to the latter. The relationship among the three spectra is studied and it is shown that whenever these secondary spectra are disjoint then an invariant complementary bundle can be found. The results have implications in the theory of perturbation of invariant manifolds. The problem is studied in the setting of skew-product dynamical systems and the results are applicable to block triangular systems of ordinary differential equations.     

Generalized Gram Matrix and Its Application to the Observability Problem for Linear Nonsteady Systems

Sufficient conditions for the nondegeneracy of a generalized Gram matrix are obtained. In particular, it is shown that the generalized Gram matrix is nondegenerate for the Chebyshev systems of functions. An application of the results to the observability problems for linear nonsteady systems of ordinary differential equations are given. In terms of the observability matrix, necessary and sufficient conditions of the complete and total observability by means of finite-parameter solving operations are established.     

Application of the method of special series to the representation of solutions of the Lin-Reissner-Tsien equation

For the two-dimensional Lin-Reissner-Tsien equation, which describes nonstationary gas flows, we construct new classes of solutions with functional arbitrariness in the form of series in powers of specially chosen functions. Coefficients of such series are found successively as solutions of linear ordinary differential equations or as solutions of linear partial differential equations. The use of special series whose coefficients are determined by linear partial differential equations allowed us to satisfy two given additional boundary conditions exactly. For one class of flows, these coefficients were found in an explicit form from linear equations of the hyperbolic type; for another one, they were found from linear equations of the parabolic type. This circumstance was used to prove the convergence of such series and to study the asymptotics of the solutions constructed. We present results of numerical calculations on nonstationary transonic flow around a wedge.     

Admissible Domains for Higher Order Differential Equations

This paper analyzes the geometric structure of certain domains in the complex plane which arise in the asymptotic theory of linear ordinary differential equations containing a parameter. These domains, called admissible, are domains in which an asymptotic representation of the solution of the differential equation may be found and across whose boundaries these representations may undergo a rapid change of asymptotic behavior (the Stokes phenomenon). A knowledge of the disposition of those domains associated with a particular differential equation is necessary for a satisfactory asymptotic theory of the equation. The main analysis gives necessary and sufficient conditions for identifying admissible domains and gives a procedure for obtaining particular admissible subdomains of a given domain. Sufficient conditions are established to determine the maximality of admissible domains. A section of examples is included to highlight the salient features of this theory. In all of the results, criteria involving only purely local properties of the boundary are needed to determine the global properties of admissibility and maximal admissibility .     

Bifurcation of periodic solutions in differential inclusions

Ordinary differential inclusions depending on small parameters are considered such that the unperturbed inclusions are ordinary differential equations possessing manifolds of periodic solutions. Sufficient conditions are determined for the persistence of some of these periodic solutions after multivalued perturbations. Applications are given to dry friction problems.     

Linear differential-algebraic equations perturbed by Volterra integral operators

We study linear inhomogeneous vector ordinary differential equations of arbitrary order in which the matrix multiplying the highest derivative of the unknown vector function is singular in the domain where the equations are defined. We also study perturbations (not necessarily small) of such equations, which are linear integro-differential equations with a Volterra operator. We obtain sufficient conditions for the solvability of such equations and give representations of their general solutions; solvability and uniqueness conditions are also given for initial value problems for such equations. The influence of small perturbations of the free term and the initial data on the solution is considered. A numerical method is suggested. The results of numerical experiments are given.     

Observability of nonlinear systems

New necessary and sufficient conditions are presented for the observability of systems described by nonlinear ordinary differential equations with nonlinear observations. The conditions are based on extension of the necessary and sufficient conditions for observability of time-varying linear systems to the linearized trajectory of the nonlinear system. The result is that the local observability of any initial condition can be readily determined, and the observability of the entire initial domain can be computed. The observability of constant parameters appearing in the differential equations is also considered. Examples are presented to illustrate the theory.This research was supported by the National Science Foundation, Grant No. NSF GK-10136.     

WEIGHTED PSEUDO ALMOST-PERIODIC SOLUTIONS OF SHUNTING INHIBITORY CELLULAR NEURAL NETWORKS WITH MIXED DELAYS

In this paper,we prove the existence and the global exponential stability of the unique weighted pseudo almost-periodic solution of shunting inhibitory cellular neural networks with mixed time-varying delays comprising different discrete and distributed time delays.Some sufficient conditions are given for the existence and the global exponential stability of the weighted pseudo almost-periodic solution by employing fixed point theorem and differential inequality techniques.The results of this paper complement the previously known ones.Finally,an illustrative example is given to demonstrate the effectiveness of our results.     

高阶常微分方程的振动性和渐进性

我们考虑一类高阶常微分方程,给出了方程的解的振动性和渐进性的充分条件,所得结果改进和推广了相关文献中的结果。     

Renormalized solutions of some transport equations with partially <Emphasis Type="Italic">W</Emphasis><Superscript>1,1</Superscript> velocities and applications

We prove existence and uniqueness of renormalized solutions of some transport equations with a vector field that is not W^1,1 with respect to all variables but is of a particular form. Two specific applications of this new result are then treated, based upon the equivalence between transport equations and ordinary differential equations. The first one consists of a result about the dependance upon initial conditions for solutions of ODEs. The second one is related to some stochastic differential equations arising in the modelling of polymeric fluid flows.     

Oscillation criteria for a forced mixed type Emden-Fowler equation with impulses

Some oscillation criteria for a forced mixed type Emden-Fowler equation with impulses are given. When the impulses are dropped, our results extend those of Sun and Meng [Y.G. Sun, F.W. Meng, Interval criteria for oscillation of second-order differential equations with mixed nonlinearities, Appl. Math. Comput. 15 (2008) 375-381], Sun and Wong [Y.G. Sun, J.S.W. Wong, Oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities, J. Math. Anal. Appl. 334 (2007) 549-560] for second-order forced ordinary differential equation with mixed nonlinearities, Nasr [A.H. Nasr, Sufficient conditions for the oscillation of forced superlinear second order differential equations with oscillatory potential, Proc. Am. Math. Soc. 126 (1998) 123-125], Yang [Q. Yang, Interval oscillation criteria for a forced second order nonlinear ordinary differential equations with oscillatory potential, Appl. Math. Comput. 135 (2003) 49-64] for forced superlinear Emden-Fowler equation, Kong [Q. Kong, Interval criteria for oscillation of second-order linear differential equations, J. Math. Anal. Appl. 229 (1999) 483-492] for unforced second order linear differential equations, and Wong [J.S.W. Wong, Oscillation criteria for a forced second order linear differential equation, J. Math. Anal. Appl. 231 (1999) 235-240] for forced second order linear differential equation.