On the structure of the regions of stability of linear canonical systems of differential equations with periodic coefficients

Gel′fand, I. M. and Lidskiĭ

doi:10.1090/trans2/008/06

Cited by 56 publications

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“…However our treatment differs somewhat from his and the results are fundamental for what follows. In § 3 we carry over to canonical systems the elegant theory of stability domains due to Gelfand and Lidskii [3]. In § 4 we prove perturbation theorems analogous to those obtained by Diliberto [2] in the Hamiltonian case.…”

Section: Introductionmentioning

confidence: 90%

“…where I k denotes the k x k unit matrix. In recent years the stability properties of Hamiltonian systems whose coefficient matrix H(t) is periodic have been deeply investigated, mainly by Russian authors ([2], [3], [5], [7]). An excellent survey of the literature is given in [6].…”

Section: Introductionmentioning

confidence: 99%

“…In studying the canonical system (1) there is no loss of generality in supposing that THEOREM 1. The system (3) is stable if and only if its monodromy matrix is stable.…”

Section: Introductionmentioning

confidence: 99%

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On the stability of linear canonical systems with periodic coefficients

Coppel

Howe

1965

J. Aust. Math. Soc.

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By a linear canonical system we mean a system of linear differential equations of the formwhereJis an invertible skew-Hermitian matrix andH(t) is a continuous Hermitian matrix valued function. We reserve the name Hami1tonia for real canonical systems withwhereIkdenotes thek×kunit matrix. In recent years the stability properties of Hamiltonian systems whose coefficient matrixH(t) is periodic have been deeply investigated, mainly by Russian authors ([2], [3], [5], [7]). An excellent survey of the literature is given in [6]. The purpose of the present paper is to extend this theory to canonical systems. The only work which we know of in this direction is a paper by Yakubovič [9].

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Section: Introductionmentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

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On the stability of linear canonical systems with periodic coefficients

Coppel

Howe

1965

J. Aust. Math. Soc.

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“…Rappelons quelques elements de la théorie de Krein (voir [23 ], [19 ], et surtout [29 ], chapitre III). Soit 03C9 e C, avec 1, un multiplicateur caracteristique d'ordre m, c'est-a-dire une valeur propre d'ordre m de R(T).…”

Section: Ekelandunclassified

“…Les resultats precedents permettent aussi d'obtenir des lumieres sur la stabilite des solutions periodiques de stabilite forte n'a de sens que pour les matrices symplectiques : elle signifie alors que la matrice est stable, et qu'il n'y a pas de valeurs propres multiples de type ( p, q) avec p 0 (voir [19], [23] ] ou [29]; on a défini le type au § IV). En d'autres termes, les valeurs propres multiples sont de type (p, 0) (positif) ou (0, q) (negatif).…”

Section: Complements -unclassified

Une théorie de Morse pour les systèmes hamiltoniens convexes

Ekeland

1984

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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L'accès aux archives de la revue « Annales de l'I. H. P., section C » (http://www.elsevier.com/locate/anihpc) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

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The Role of the Legendre Transform in the Study of the Floer Complex of Cotangent Bundles

Abbondandolo

Schwarz

2014

Comm Pure Appl Math

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Consider a classical Hamiltonian H on the cotangent bundle T * M of a closed orientable manifold M , and let L : T M → R be its Legendre-dual Lagrangian. In a previous paper we constructed an isomorphism Φ from the Morse complex of the Lagrangian action functional which is associated to L to the Floer complex which is determined by H. In this paper we give an explicit construction of a homotopy inverse Ψ of Φ. Contrary to other previously defined maps going in the same direction, Ψ is an isomorphism at the chain level and preserves the action filtration. Its definition is based on counting Floer trajectories on the negative half-cylinder which on the boundary satisfy "half" of the Hamilton equations. Albeit not of Lagrangian type, such a boundary condition defines Fredholm operators with good compactness properties. We also present a heuristic argument which, independently on any Fredholm and compactness analysis, explains why the spaces of maps which are used in the definition of Φ and Ψ are the natural ones. The Legendre transform plays a crucial role both in our rigorous and in our heuristic arguments. We treat with some detail the delicate issue of orientations and show that the homology of the Floer complex is isomorphic to the singular homology of the loop space of M with a system of local coefficients, which is defined by the pull-back of the second Stiefel-Whitney class of T M on 2-tori in M .

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On the structure of the regions of stability of linear canonical systems of differential equations with periodic coefficients

Cited by 56 publications

References 0 publications

On the stability of linear canonical systems with periodic coefficients

On the stability of linear canonical systems with periodic coefficients

Une théorie de Morse pour les systèmes hamiltoniens convexes

The Role of the Legendre Transform in the Study of the Floer Complex of Cotangent Bundles

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