Sigrun Bodine

Asymptotic diagonalization of linear difference equations

Our purpose is to obtain conditions under which a kinematic similarity exists that reduces the linear difference equation to a linear equation with coefficient matrix B(n) being diagonal for sufficiently large n. It is shown that if A(n) is the sum of a bounded sequence plus a sequence as and if the omega—limit set of , in the associated skew—product flow, has full spectrum then such a kinematic similarity exists.

Exponentially asymptotically constant systems of difference equations with an application to hyperbolic equilibria

In a recent paper, Agarwal and Pituk have considered scalar linear difference equations whose coefficients are asymptotically constant and whose corresponding perturbations are exponentially small. Using the method of generating functions and results from complex analysis, they derived an asymptotic representation for solutions, which was then applied to study the asymptotic behaviour of solutions of certain nonlinear autonomous scalar difference equations near a hyperbolic equilibrium. Here, we first show using standard matrix analysis how their results can be extended to systems of linear difference equations and that the error estimates can be made more precise. Our method also can be extended to weakly nonlinear systems. That result can in turn be used to analyze solutions of some autonomous nonlinear difference systems near an equilibrium.

A Dynamical Systems Result On Asymptotic Integration Of Linear Differential Systems

Abstract We are interested in the asymptotic integration of linear differential systems of the form x ′=[ Λ ( t )+ R ( t )] x , where Λ is diagonal and R ∈ L p [ t 0 ,∞) for p ∈[1,2]. Our dichotomy condition is in terms of the spectrum of the omega-limit set ω Λ . Our results include examples that are not covered by the Hartman–Wintner theorem.

On the Summability of Formal Solutions in Liouville-Green Theory

We consider the second-order differential equation ∈2y″ = (1+∈2ψ(x, ∈))y with a small parameter ∈, where ψ is even with respect to ∈. It is well known that it has two formal solutions y±(x, ∈) = e±x/∈h±(x, ∈), where h±(x, ∈) is a formal series in powers of ∈ whose coefficients are functions of x.It has been shown [4] that one resp. both of these solutions are 1-summable in certain directions if ψ satisfies certain conditions, in particular, concerning its x-domain. In the present article we give necessary (and sufficient) conditions for 1-summability of one or both of the above formal solutions in terms of ψ.The method of proof involves a certain inverse problem, i.e., the construction of a differential equation of the above form exhibiting a prescribed Stokes phenomenon with respect to ∈.

Asymptotic integration of differential and difference equations

Introduction, Notation, and Background.- Asymptotic Integration for Differential Systems.- Asymptotics for Solutions of Difference Systems.- Conditioning Transformations for Differential Systems.- Conditioning Transformations for Difference Systems.- Perturbations of Jordan Differential Systems.- Perturbations of Jordan Difference Systems.- Applications to Classes of Scalar Linear Differential Equations.- Applications to Classes of Scalar Linear Difference Equations.- Asymptotics for Dynamic Equations on Time Scales.

A Hartman–Wintner theorem for dynamic equations on time scales

Classical results of the Hartman–Wintner type lead to the asymptotic integration of certain types of linear, non-autonomous systems of differential or difference equations. Here, we show how to extend such results to linear dynamic equations on time scales having bounded graininess, using both averaged as well as pointwise dichotomy conditions.

Perturbations of Jordan Differential Systems

Whereas in Chaps. 2 and 4, we studied the asymptotic behavior of solutions of perturbations of diagonal systems of differential equations, we are now interested in the asymptotic behavior of solutions of systems of the form

Conditioning Transformations for Difference Systems

\displaystyle{ y^{{\prime}} = \left [J(t) + R(t)\right ]y(t)t \geq t_{ 0}, }