Classification of linear time-varying difference equations under kinematic similarity

Gohberg, Israel; Kaashoek, M. A.; Kos, J.

doi:10.1007/bf01203027

Cited by 25 publications

(17 citation statements)

References 9 publications

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“…Remark This chain of reasoning not only applies to p = 4, but also for aij p > O iying in the compiement of ji, 31 u [5,7], This impiies that …”

Section: Examplementioning

confidence: 93%

“…Since the distance between two consecutive elements la, -a,+ll 5 2 / ( n + I), there is also a subsequence converging to c and the constant matrix sequence A(') with each e!ernent being diag(2 + c, 6 + c) is an element in the omega-limit set. Thus [l,3] U [5,7] is in the spectrum of w~.…”

Section: Examplementioning

confidence: 98%

“…The change of variables x(n> = Q(n)y(n) then transforrrs (1.1) into (1,2). Kinematic similarity was studied by Markus [7] in the context of ordinary linear differential equations and was later modified for difference equations (see, for example, [5]). …”

Section: Introductionmentioning

confidence: 99%

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Asymptotic diagonalization of linear difference equations^*

Bodine¹,

Sacker

2001

Journal of Difference Equations and Applications

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lowiial of Dkffeicrence Equnrions mid Applicatio~is iLiO!, Yai 7 , pp 03; -656 Repnnts nvailahle directly from the puhlichei Photocopy~ng permitted b) license only .; 2001 OPA iOverseas Publishers .Associ?rion) N V Pubiishd by iiicilse uniisi [he Gnriinr? snd RrcachOur purpose is to obtain conditions under which a kinematic similarity exists that reduces the linear difl'erence equation x(n+ 1) = A(n)x(n) 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 A(n) plus a sequence P(n) --to as n-oo and if the omega-limit set of A (nj, in the associated skew-product flow, has full spectrum then such a kinematic similarity exists.

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“…Remark This chain of reasoning not only applies to p = 4, but also for aij p > O iying in the compiement of ji, 31 u [5,7], This impiies that …”

Section: Examplementioning

confidence: 93%

Section: Examplementioning

confidence: 98%

See 1 more Smart Citation

Asymptotic diagonalization of linear difference equations^*

Bodine¹,

Sacker

2001

Journal of Difference Equations and Applications

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“…An identical description of the set σ(T A ) using local spectral theory can be found in [ the single value extension property (SVEP) is given in [11]. Moreover, in [18] the similarity of weighted shifts T A is discussed for several sequences A k . The previously described shape of the spectrum changes when dealing with unilateral shifts.…”

Section: Nonautonomous Hyperbolicitymentioning

confidence: 99%

Fine Structure of the Dichotomy Spectrum

Pötzsche

2012

Integr. Equ. Oper. Theory

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The dichotomy spectrum is a crucial notion in the theory of dynamical systems, since it contains information on stability and robustness properties. However, recent applications in nonautonomous bifurcation theory showed that a detailed insight into the fine structure of this spectral notion is necessary. On this basis, we explore a helpful connection between the dichotomy spectrum and operator theory. It relates the asymptotic behavior of linear nonautonomous difference equations to the point, surjectivity and Fredholm spectra of weighted shifts. This link yields several dynamically meaningful subsets of the dichotomy spectrum, which not only allows to classify and detect bifurcations, but also simplifies proofs for results on the long term behavior of difference equations with explicitly time-dependent right-hand side.

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“…Markus [17] introduced the notion of kinematic similarity in the set of all n × n continuous matrices defined on [t 0 , ∞) and showed that the relationship of kinematic similarity is an equivalence relation preserving the type numbers of the linear differential systems. Gohberg et al [14] studied the problem to classify linear time-varying systems of difference equations under kinematic similarity. Conti [12] introduced the concept of t ∞ -similarity in the set of all n × n continuous matrices defined on R + = [0, ∞) and showed that t ∞ -similarity is an equivalence relation preserving strict, uniform and exponential stability of linear homogeneous differential systems.…”

Section: Introductionmentioning

confidence: 99%