The Sufficiency of the Matkowsky Condition in the Problem of Resonance

“…En particulier il n'est pas possible de considérer la somme usuelle de la série y car celle-ci est en général divergente. Ce résultat trouve une application dans le problème de la résonance d'Ackerberg-O'Malley [37,30].…”

Section: Equations Analytiques Par Rapport à εunclassified

Sur le retard à la bifurcation

Fruchard

Schäfke

2008

Revue Africaine De Recherche en Informatique Et Mathématiques Appliquées

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International audience We give a non-exhaustive overview of the problem of bifurcation delay from its appearance in France at the end of the 80ies to the most recent contributions. We present the bifurcation delay for differential equations as well as for discrete dynamical systems. Nous donnons un aperçu non exhaustif du problème du retard à la bifurcation, depuis son apparition en France à la fin des années 1980 jusqu’aux contributions les plus récentes. Le problème et les résultats sont présentés d’une part pour les équations différentielles et d’autre part pour les systèmes dynamiques discrets

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Phragmén-Lindelöf theorem in a cohomological form

Lin¹

1983

Proc. Amer. Math. Soc.

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Abstract.The main result of this paper is as follows. Given functions <#>,(f).... ,r>,,(e) which are holomorphic in sectors S.S", respectively, where S¡ U ■ ■ • US, = {c: |argE|<7r/2a, 0 <|e|< p} for a > 1, p > 0, set ,A = $, -4>k if S, n Sk ¥= 0. Then {4>/k} satisfy cocycle conditions /k + kl = ;/ whenever S; PI Sk nS,^ 0. In addition to the conditions |J< MQ and \r\< M0 on the two rays of the boundary (i.e. arg e = w/2 a ), and |^;(e)|< A exp(c/|e|) in S; for some positive numbers A and c, j = l,2,...,c, if the {<¡>;} satisfy the conditions |,|< M on S,, 7 = 1,2.c (From the cohomological point of view, we can get global results for ;, once the local data on cocycles is known.)1. Introduction. Let ß = (e: -it/2a < arg e < w/2a, 0 <|e|< p} be a sector in the right half complex e-plane, where a > 1 and p is a positive number. Let / be a complex valued function which is continuous on ß* = {e: -7r/2a < arg e < ir/2a, 0 <|e|=s p}, holomorphic in ß, and there are positive constants M and c such that |/(e) |< Mexp(c/|e |) for all e G ß. Assume, furthermore, that |/(e) |< M for all e on the boundary of ß. Then, the Phragmen-Lindelöf theorem states that |/(e)|=£ M for all e in ß (see [2, p. 282]). In this paper, we shall generalize this theorem in a cohomological form; i.e. given functions <£,(£),... ,"(e) which are holomorphic in sectors S,,... ,S", respectively, ß = Sx U • • ■ U 5"", set k if Sj C\ Sk¥= 0. Then {,*} satisfy cocycle conditions k/ = d>7 whenever S, D Sk C\ S¡ ¥^ 0. With this property, our theorem can be stated in the following way: If the {<£•} satisfy the conditions |" |< M0 on the two rays of the boundary (i.e. arg e = ±tr/2a), then we get I.|< M on Sj,j = l,...,v (cf. Theorem 1). From the cohomological point of view, we can get global results for

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The Sufficiency of the Matkowsky Condition in the Problem of Resonance

Cited by 3 publications

References 6 publications

A survey of some results on overstability and bifurcation delay

A survey of some results on overstability and bifurcation delay

Sur le retard à la bifurcation

Phragmén-Lindelöf theorem in a cohomological form

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