Functional equations and conjugacy of local diffeomorphisms of a finite smoothness class

Belitskii, Genrich

doi:10.1007/bf01075731

Cited by 89 publications

(90 citation statements)

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“…On en déduit queã n représente le mêmeélément que a n (et donc que a n ) dans π 1 (R 2 \ {(0, 0)}) = Z. D'autre partã n est formé d'un segment de feuille de Lα et d'un segment deD 1 . NotonsH l'holonomie de Lα (orienté dans le sens trigonométrique direct) surD 1 .…”

Section: 4unclassified

“…D'autre partã n est formé d'un segment de feuille de Lα et d'un segment deD 1 . NotonsH l'holonomie de Lα (orienté dans le sens trigonométrique direct) surD 1 . On en déduit quex n =H n (x 0 ) si α < 0 etx n =H −n (x 0 ) si α > 0.…”

Section: 4unclassified

“…On en déduit quex n =H n (x 0 ) si α < 0 etx n =H −n (x 0 ) si α > 0. D'autre part, d'après le Lemme 4.14,x n est plus proche de (0, 0) surD 1 …”

Section: 4unclassified

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Équivalence topologique de connexions de selles en dimension 3

Bonatti¹,

Dufraine

2003

Ergod. Th. Dynam. Sys.

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Résumé. Nous donnons des invariants complets pour l'équivalence topologique de champs de vecteurs, en dimension 3, au voisinage d'une connexion (par des variétés invariantes de dimension 1) entre des selles possédant des valeurs propres complexes.Abstract. We give a complete set of invariants for the topological equivalence of vector fields on 3-manifolds in the neighborhood of a connection by one-dimensional separatrices between two hyperbolic saddles having complex eigenvalues.More precisely, let X be a C 2 vector field on a 3-manifold, having two hyperbolic zeros p, q of saddle type, such that p admits a contracting complex eigenvalue −c p (1 + iα), c p > 0, α = 0, and q admits an expanding complex eigenvalue c q (1 + iβ), c q > 0, β = 0. We assume that a one-dimensional unstable separatrix of p coincides with a onedimensional stable separatrix of q, and we call the connection the compact segment γ consisting of p, q and their common separatrix (see Figure 1). Such a connection is a codimension two phenomenon.The behaviour of a vector field X in the neighborhood of the connection is given, up to topological conjugacy, by the linear part of X in the neighborhood of p and q and by the transition map between two discs transversal to X in those neighborhoods. First, we choose coordinates in the neighborhoods of p and q in order to put X in canonical form and we show that, up to topological equivalence, we can assume that the transition map is linear. Then our main result is as follows :• When the linear transition map (expressed in the chosen coordinates) is conformal or when its modulus of conformality is small (i.e. less than some function ψ(α, β)), there is no topological invariant : every two such vector fields are topologically equivalent. •On the other hand, when the modulus of conformality is greater than ψ(α, β), there are two topological invariants : one is equal to the ratio α/β, the other one is related to the modulus of conformality of the transition map. Equivalence topologique de connexions de selles 1349Le comportement (àéquivalence topologique près) d'un champ au voisinage d'une connexion est essentiellement décrit par sa partie linéaire au voisinage des deux points selles, et par une transition T , qui est l'holonomie du champ le long de la connexion entre deux disques transverses au champ et contenus dans des ouverts de linéarisation au voisinage des points singuliers.Dans le cas où les valeurs propres associées aux points p et q sont toutes réelles, Vegter (voir [11, 12]) montre que, génériquement, il n'y a pas de module de stabilité : toute perturbation du champ X, suffisamment C 1 -petite et préservant la connexion, est topologiquementéquivalenteà X au voisinage de la connexion. Ce résultat aété généralisé en toutes dimensions par Dias Carneiro et Palis (voir [4]).Nous considérons ici le cas où les points selles p et q possèdent chacun une valeur propre complexe (non-réelle), stable pour p (donc de partie réelle négative) et instable pour q (de partie réelle positive). Nous donnons dans ce c...

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Section: 4unclassified

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Équivalence topologique de connexions de selles en dimension 3

Bonatti¹,

Dufraine

2003

Ergod. Th. Dynam. Sys.

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“…The proof of the theorem depends on a C 1 version of the HartmanGrobman theorem. More detailed results concerning smooth equivalence can be found in [6], [19], [21], [22]. In general, a C 1 hyperbolic map is only topologically conjugate equivalent to the linear part in a small neighborhood of the fixed point, thus Lemma 3.1 no longer holds for C 1 maps near the hyperbolic fixed point without assuming that f is contracting.…”

Section: Hongbin Chen and Yi LImentioning

confidence: 99%

Existence, uniqueness, and stability of periodic solutions of an equation of duffing type

Hong-bin¹,

Li²

2007

Discrete &Amp; Continuous Dynamical Systems - A

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Abstract. We consider a second-order equation of Duffing type. Bounds for the derivative of the restoring force are given which ensure the existence and uniqueness of a periodic solution. Furthermore, the unique periodic solution is asymptotically stable with sharp rate of exponential decay. In particular, for a restoring term independent of the variable t, a necessary and sufficient condition is obtained which guarantees the existence and uniqueness of a periodic solution that is stable. §1. Introduction. This paper is devoted to the existence, uniqueness and stability of periodic solutions of the Duffing-type equationwhere g(t, x) is a T -periodic function in t and h(t) is a T -periodic function. The existence and multiplicity of periodic solutions of (1.1) or more general types of nonlinear second-order differential equations have been investigated extensively by many authors since C. Fabry, J. Mawhin and M.N. Nkashama initiated the study of the Ambrosetti-Prodi problem with periodic boundary condition [7]. However, the stability of periodic solutions is less extensively studied. In [18] R. Ortega studied (1.1) from a stability point of view and obtained an Ambrosetti-Proditype theorem under an assumption of convex nonlinearity. A.C. Lazer and P.J. McKenna established stability results by converting the equation (1.1) to a fixedpoint problem [14]. Recently, more complete results concerning the stability of periodic solutions of (1.1) were obtained by J.M. Alonso and R. Ortega [1,2]. Under the condition that the derivative of the restoring force is independent of t and positive, they found sharp bounds that guarantee global asymptotic stability. In [1], optimal bounds for stability are obtained. But the above results do not cover our Theorem 1, since in the theorem the derivative of the restoring force may be negative 2000 Mathematics Subject Classification. 34C25, 34C10.

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“…We further assume that coordinates are chosen so that the local stable manifold of the origin lies on the x-axis and the local unstable manifold of the origin lies on the y-axis. It follows from 3.1 and Samovol's version of Sternberg's linearization theorem (Samoval 1972, Belickii 1973) that (provided the vector field is at least C 7 ) we can make a C 3 -local change of coordinates so that the vector field is linear near the origin:…”

Section: Attractors For Product Systemsmentioning

confidence: 99%

Product dynamics for homoclinic attractors

Ashwin

Field

2005

Proc. R. Soc. A.

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Heteroclinic cycles may occur as structurally stable asymptotically stable attractors if there are invariant subspaces or symmetries of a dynamical system. Even for cycles between equilibria, it may be difficult to obtain results on the generic behaviour of trajectories converging to the cycle. For more complicated cycles between chaotic sets, the nontrivial dynamics of the 'nodes' can interact with that of the 'connections'. This paper focuses on some of the simplest problems for such dynamics where there are direct products of an attracting homoclinic cycle with various types of dynamics. Using a precise analytic description of a general planar homoclinic attractor, we are able to obtain a number of results for direct product systems.We show that for flows that are a product of a homoclinic attractor and a periodic orbit or a mixing hyperbolic attractor, the product of the attractors is a minimal Milnor attractor for the product. On the other hand, we present evidence to show that for the product of two homoclinic attractors, typically only a small subset of the product of the attractors is an attractor for the product system.

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Functional equations and conjugacy of local diffeomorphisms of a finite smoothness class

Cited by 89 publications

References 4 publications

Équivalence topologique de connexions de selles en dimension 3

Équivalence topologique de connexions de selles en dimension 3

Existence, uniqueness, and stability of periodic solutions of an equation of duffing type

Product dynamics for homoclinic attractors

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