Topological equivalence of normally hyperbolic dynamical systems

Palis, Jacob; Takens, Floris

doi:10.1016/0040-9383(77)90040-4

Cited by 71 publications

(34 citation statements)

References 8 publications

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“…Remark 1. An alternative proof of Theorem 3.2 can be constructed by first proving the existence of a collection of smooth fibers which are transversal to every center manifold and which are transported by the flow of (3.11), while the flow in each fiber is exponentially attracting for t -* + oo with approximately the rate prescribed by the eigenvalues of L2 [4,8,12,34,36]. For t -* -oo the flow in the fibers is expanding and for solutions which remain bounded for all negative t this implies that the corresponding fibers intersect with at most one center manifold.…”

mentioning

confidence: 99%

Properties of Center Manifolds

Sijbrand¹

1985

Transactions of the American Mathematical Society

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Abstract.The center manifold has a number of puzzling properties associated with the basic questions of existence, uniqueness, differentiability and analyticity which may cloud its profitable application in e.g. bifurcation theory. This paper aims to deal with some of these subtle properties.Regarding existence and uniqueness, it is shown that the cut-off function appearing in the usual existence proof is responsible for the selection of a single center manifold, thereby hiding the inherent nonuniqueness. Conditions are given for different center manifolds at an equilibrium point to have a nonempty intersection. This intersection will include at least the families of stationary and periodic solutions crossing through the equilibrium. In the case of nonuniqueness the difference between any two center manifolds will be less than c, exp(c2A_1) with c, and c2 constants, which explains why the formal Taylor expansions of different center manifolds are the same, while the expansions do not converge.The differentiability of a center manifold will in certain cases decrease when moving out of the origin and a simple example shows how the differentiability may be lost.Center manifolds are usually not analytic; however, an analytic manifold may exist which contains all periodic solutions of a certain type but which may otherwise not be invariant. Using this manifold, a new and very simple proof of the Lyapunov subcenter theorem is given.

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confidence: 99%

Properties of Center Manifolds

Sijbrand¹

1985

Transactions of the American Mathematical Society

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“…Un difféomorphisme ^GDiff"' 8 est dit k-déterminé pour la (S^-conjugaison (pour 0 < s < k < £) si tout g' € Diff"' 6 ayant (*) Le second des auteurs a profité d'un séjour à 11MPA de Rio de Janeiro en août-septembre 1980 pour achever la rédaction de cet article. même h-jei que g en 0 est (î^-conjugé à g. De même danŝ "' Ê , on peut définir la notion de A-détermination pour la ô 5 -conjugaison et la (^-équivalence (où 0 < s < k < £).…”

Section: Germes De Difféomorphismes Et De Champs De Vecteurs En Classunclassified

“…sont -équivalents, il en est de même de X et Y, pourvu que leurs espaces stables aient même dimension [8]. Dans certains cas, le résultat de Takens sur les champs partiellement hyperboliques [10], permet même de montrer que si Xç,Y^, sont (5^-conjugés, pour 0<fe<£ (î: différentiabilité de W^ et WÇ), alors X et Y sont aussi <° ^-conjugués.…”

Section: Germes De Difféomorphismes Et De Champs De Vecteurs En Classunclassified

“…fonction définie dans le théorème V.2 ; considérons Wx , une variété centrale de classe > K. Le théorème V.2 implique que la restriction X de X à Wy est Mais maintenant, une équivalence topologique entre X et Y, restriction de deux germes de champs à des variétés centrales, se relève en une équivalence entre les germes X et Y (voir [8]). Ceci achève la démonstration du théorème V. 1.…”

Section: Germes De Diffeomorphismes Et De Champs De Vecteursunclassified

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Germes de difféomorphismes et de champs de vecteurs en classe de différentiabilité finie

Dumortier¹,

Roussarie²

1983

Annales De L’institut Fourier

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“…In [10], it is considered the problem of extending topological equivalence defined on normally invariant submanifolds to neighborhoods of them. We shall work with the same plan in mind.…”

Section: Introductionmentioning

confidence: 99%