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International audience; This paper addresses the numerical computation of periodic solutions of nonlinear differential systems involving fractional derivatives. For this purpose, the Harmonic Balance Method and the Asymptotic Numerical Method are combined, generalizing an approach largely followed in non-fractional systems. This enables to perform the continuation of periodic solutions of fractional systems with respect to a system parameter or to the fractional order. In the particular case of a constant fractional order, the results are validated by a successful comparison with an alternative formulation based on the diffusive representation of fractional operators. The new numerical strategy presented here allows to simulate phenomena still lacking from theoretical foundations. For example, the numerical experiments proposed here lead to a bifurcation similar to the Hopf bifurcation, well known in the case of non-fractional systems. Throughout this article, the Weyl derivative is used; its link with the classic Caputo derivative is elucidated.

International audience; Time-periodic solutions of dynamical systems can be looked for using a discretization method. This paper tests the Harmonic Balance Method (HBM) on a one-degree-of-freedom system (mass, damper, spring, belt) with a regularized friction law. Its relative error is computed with respect to the number of discretization unknowns. Despite the widespread idea that frequency methods are hardly applicable to friction problems, the HBM compares well with a classical time-domain method for this nonlinear system. The main conclusion of this article is that the HBM, without any specific optimization, is well suited for regularized friction.

International audience; Periodic solutions of systems with friction are difficult to investigate because of the irregular nature of friction laws. This paper examines periodic solutions and most notably stick-slip, on a simple one-degre-of-freedom system (mass, spring, damper, belt), with Coulomb's friction law, and with a regularized friction law (i.e. the friction coefficient becomes a function of relative speed, with a stiffness parameter). With Coulomb's law, the stick-slip solution is constructed step by step, which gives a usable existence condition. With the regularized law, the Asymptotic Numerical Method and the Harmonic Balance Method provide bifurcation diagrams with respect to the belt speed or normal force, and for several values of the regularization parameter. Formulations from the Coulomb case give the means of a comparison between regularized solutions and a standard reference. With an appropriate definition, regularized stick-slip motion exists, its amplitude increases with respect to the belt speed and its pulsation decreases with respect to the normal force.

International audience; Nonlinear systems can have periodic solutions evolving with the parameters of the system. Studying this evolution (numerical continuation of solutions) uncovers sought-after regimes in musical acoustics : many musical instruments rely on auto-oscillation, that is, the excitation of a nonlinear system coupled with a linear resonator, where some parameters may be adjusted by the player. Periodic solutions can be approximated as truncated Fourier series (Harmonic Balance Method) ; the period is one of the unknowns. Several stable or unstable solutions can be found for the same playing parameters thanks to continuation. An important challenge is the continuation of quasi-periodic solutions, also called multiphonic sounds by musicians. Depending on the context, these oscillation regimes are considered pleasant (jazz or contemporary music for instance) or unpleasant (classical music). We developed a method based on double Fourier series, coupled with a continuation technique. The two base frequencies are unknowns and incommensurable. The system is reformulated as quadratic in order to allow straight interface with previous work on periodic harmonic balance. This method is illustrated on simple models relevant to musical acoustics, though the method can be applied to many nonlinear problems, without a priori knowledge of the solutions.

National audience; Un système non linéaire peut avoir des solutions périodiques , qui évoluent selon les paramètres du système. L'étude de cette évolution des solutions, appelée continuation, a un intérêt tout particulier en acoustique musicale, de nombreux instruments reposant sur l'excitation d'un système non linéaire dont le musicien contrôle certains paramètres. Les solutions périodiques peuvent être cherchées sous forme de séries de Fourier tronquées (méthode de l' Équilibrage Harmonique) ; la période est une des inconnues. Plusieurs solutions stables peuvent être mises en évidence par la continuation pour les mêmes paramètres de jeu, illustrant la dépendance aux conditions initiales. La méthode donne aussi accès aux solutions instables des modèles. Un défi important concerne la recherche des solutions quasi-périodiques à deux fréquences incommensurables (aussi appelés "sons multiphoniques" par les musiciens). Nous avons développé une méthode de recherche de ces solutions, représentées par des séries doubles de Fourier. Comme dans le cas périodique , une méthode de continuation permet de suivre la branche de solutions obtenues. Nous illustrons cette méthode sur un modèle réduit d'instrument à anche.

Under certain hypothesises, we prove that a map which is an isometry for the Caratheodory infinitesimal metric at a point is an analytic isomorphism onto its image., 6 pages

Results of H. Cartan about holomorphic automorphisms on bounded domains are generalized to the case of hyperbolic manifolds in the sense of Kobayashi. In this setting, we give an identity theorem together with its topological version. We show also that a sequence of automorphisms which converges uniformly on some nonempty open set, has a limit on the whole space which is an automorphism. At the end of the paper, conditions are given for the sequence of iterates of a self holomorphic map in order to be an automorphism.

We study discrete fixed point sets of holomorphic self-maps of complex manifolds. The main attention is focused on the cardinality of this set and its configuration. As a consequence of one of our observations, a bounded domain in ${\Bbb C}^n$ with no non-trivial holomorphic retractions is constructed., 12 pages

Dans ce papier, je montre pour la distance de Caratheodory sur un produit continu B de domaines (B s ) s∈S la formule: C B (f,g) = maxC Bs (f(s),g(s)). In this paper, I prove for the Caratheodory distance on a continuous product B ofdomains (B s ) s∈S the formula: C B (f,g) = max cB s (f(s),g(s)).

Dans cet article, nous etudions les ensembles d'unicite pour le groupe Aut(D) des automorphismes analytiques d'un domaine borne D de C n (resp. pour l'ensemble H(D,D) des fonctions holomorphes de D dans lui-meme).Dans les deux cas, nous montrons qu'il existe des ensembles d'unicite contenus dans D n + 1 ; pour Aut(D), nous montrons que ces ensembles d'unicite forment un ensemble dense de D n + 1 , et pour H(D, D), que ce n'est pas le cas en general.

In this Note, we study the determination sets for the group Aut(D) of holomorphic automorphisms of a bounded domain D in Cn. To cite this article: J.-P. Vigue, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

RésuméPour un domaine convexe borné D d'un espace de Banach réflexif E, j'étudie les relations entre les propositions suivantes :(a) D est un domaine borné strictement convexe ;(b) pour tout a∈D, pour tout r>0, la boule Bc(a,r) pour la distance de Carathéodory cD est strictement convexe ;(c) pour tout a∈D, pour tout b∈D, les géodésiques complexes passant par a et b sont uniques (à composition par un automorphisme analytique de Δ près).Je montre en particulier dans cet article que (a) implique (b) et que (b) implique (c). Je montre aussi que les réciproques sont fausses.AbstractIn this paper, for a bounded convex domain D in a complex reflexive Banach space, I study the relationship between the following propositions:(a) D is a bounded strictly convex domain;(b) for every a∈D, for every r>0, the ball Bc(a,r) for the Carathéodory distance cD is strictly convex;(c) for every a and b in D, the complex geodesics ϕ :Δ→D such that a and b belong to ϕ(Δ) are unique.I prove that (a) ⇒ (b)⇒ (c). I also prove that the converses are wrong.

This paper deals with the characterization of a domain D in Cn by the Kobayashi infinitesimal metric in a neighborhood of a point a of D. I prove this characterization in the following cases: a domain D in C analytically isomorphic to the open unit disc, an hyperbolic domain D in C, a bounded strictly convex domain D in Cn and also a bounded convex domain D in Cn which is isomorphic to an open unit ball. The proofs use the result of L. Lempert on the equality of the Carathéodory and Kobayashi infinitesimal metric on convex domains and the notion of complex geodesic.RésuméThis paper deals with the characterization of a domain D in Cn by the Kobayashi infinitesimal metric in a neighborhood of a point a of D. I prove this characterization in the following cases: a domain D in C analytically isomorphic to the open unit disc, an hyperbolic domain D in C, a bounded strictly convex domain D in Cn and also a bounded convex domain D in Cn which is isomorphic to an open unit ball. The proofs use the result of L. Lempert on the equality of the Carathéodory and Kobayashi infinitesimal metric on convex domains and the notion of complex geodesic.

In this paper, I study the group of analytic automorphisms of a bounded product domain in the space C(S,C) of continuous functions on a compact space S. I prove that its automorphism group is a Lie group and I am able to prove which are the bounded symmetric ones.

RésuméLet B1 and B2 be the open unit balls of ℂn1 and ℂn2 for the norms ‖. ‖1 and ‖ . ‖2. Let f: B1 → B2 be a holomorphic mapping such that f(0) = 0. It is well known that, for every z ∈ B1, ‖f(z)‖2 ≤ ‖z‖1, and ‖f′(0)‖ ≤ 1.In this paper, I prove the converse of this result. Let f : B1 → B2 be a holomorphic mapping such that f'(0) is an isometry. If B2 is strictly convex, I prove that f(0)=0 and that f is linear. I also define the rank of a point x belonging to the boundary of B1 or B2. Under some hypotheses on the ranks, I prove that a holomorphic mapping such that f(0)=0 and that f'(0) is an isometry is linear.

In this paper we prove that a commuting family of continuous self-maps of a bounded convex domain in C n {\mathbb {C}^n} which are holomorphic in the interior has a common fixed point. The proof makes use of three basic ingredients: iteration theory of holomorphic maps, a precise description of the structure of the boundary of a convex domain, and a similar result for commuting families of self-maps of a hyperbolic domain of a compact Riemann surface.

We study fixed point sets for holomorphic automorphisms (and endomorphisms) on complex manifolds. The main object of our interest is to determine the number and configuration of fixed points that forces an automorphism (endomorphism) to be the identity. These questions have been examined in a number of papers for a bounded domain in ${\Bbb C}^n$. Here we resolve the case for a general finite dimensional hyperbolic manifold. We also show that the results for non-hyperbolic manifolds are notably different., Comment: 10 pages

An analytic space is given which is c X {c_X} -complete but not c X {c_X} -finitely compact.

An example of a two-dimensional complex space is given with the property that the continuous spectrum of the global holomorphic functions is not locally compact at any point. Introduction. The spectrum Sc(O(X)) of the global holomorphic functions 0(X) on a complex space (X, 0) is the set of all continuous complex-valued algebra homomorphisms on 0 (X) endowed with the Gelfand topology. This functional analytic concept has important applications to fundamental problems in complex analysis. For instance, due to a theorem of Igusa-Remmert-Iwahashi-Forster (see [3, 1.5]), if there is at least one point where the spectrum Sc (O(X)) is not locally compact, then the complex space (X, 0) does not have a Stein envelope of holomorphy.1 In all the examples known of complex spaces (X, 0) without a Stein envelope of holomorphy, the set of points in the spectrum Sc (O(X)) where local compactness fails is a nonempty, extremely small, proper subset of Sc (O(X)); frequently, it consists of just one point [2, 5.3; 7, 4.2, 4.3]. A natural question is whether such pathological points can be described as a thin subset of the spectrum. However, the purpose of this note is to show that the spectrum Sc (O(X)) of a complex space (X, 0) need not be locally compact at any point at all; in the example constructed here, X is two dimensional. A complex space always refers to a reduced complex space with countable topology. Construction. The example will be constructed in two steps. The main idea of the first step is to find a two-dimensional complex space (X, 0) whose spectrum Sc (O(X)) has the following description. Take a two-dimensional complex plane and attach infinitely many two-dimensional complex planes transversally along every line of a countable dense subset of lines in the first plane. In the second step, the unique position of the initial plane will be eliminated by carrying out the construction of the first step for every plane attached to the initial plane. First step of the construction. For every nonnegative integer n E N, set Sn := {(x,y) E C2: lxl < n + ?1,n < ?y < n + 1}. Let D be the Reinhardt domain in C2 obtained by removing all the sets S, n E N, from C2. Received by the editors August 19, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 32D10; Secondary 46H05. 1Whether the converse of this statement holds is open. (? 1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page

Vesentini studied complex geodesics of a bounded domain D in a complex Banach space and applied his results to the set of fixed points of an holomorphic map from D to D. In a previous paper, I studied the fixed points of an holomorphic map f from a bounded convex domain D in Cn into itself, and I proved that, given two fixed points of f, there exists a complex geodesic containing these two points and contained in the set of fixed points of f. In this paper, I prove the same result for a finite product of open unit balls in complex Hilbert spaces. The proof is more difficult, is based on a careful study of the Caratheodory distance, and uses results of Goebel, Sekowski and Stachura.

Let D D be a bounded convex domain in C n {{\mathbf {C}}^n} . We prove that the set V V of fixed points of a holomorphic map f : D → D f:D \to D is a complex submanifold of D D and, if V V is not empty, V V is a holomorphic retract of D D .

Resume Dans cet article, nous montrons le resultat suivant : Soit X une variete complexe hyperbolique pour la distance de Caratheodory et soit U un ouvert relativement compact dans X . Alors, il existe k 1 tel que, pour la metrique infinitesimale de Caratheodory, on ait E X ( x , v ) ⩽ k E U ( x , v ) . Nous obtenons aussi des resultats sur les points fixes dʼapplications holomorphes de X dans U .

We construct an analytic space X such that the Caratheodory pseudodistance cx is a true distance on X; however, cx does not define the analytic space topology of X.