ETUDE D'EQUATIONS DIFFERENTIELLES ORDINAIRES SINGULIEREMENT PERTURBEES AU VOISINAGE D'UN POINT TOURNANT PDF Download
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Author: ERIC.. MATZINGER
Publisher:
ISBN:
Category :
Languages : fr
Pages : 110
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Book Description
ON PART DE L'EQUATION DE VAN DER POL, E V DV/DU = (1U.U) V + AU, QUI EST CONNUE POUR AVOIR, POUR CERTAINES VALEURS DU PARAMETRE A (VALEUR DEPENDANT DU PARAMETRE DE PERTURBATION E QU'ON FAIT TENDRE VERS 0), DES SOLUTIONS EXCEPTIONNELLES, APPELEES CANARDS, CONTINUES ET BORNEES DANS UN VOISINAGE COMPLET DU POINT 1 ; ALORS QUE CE POINT EST HABITUELLEMENT UN POINT TOURNANT POUR LES SOLUTIONS DE CETTE EQUATION, C'EST-A-DIRE QU'ELLES SONT AU MIEUX BORNEES DANS CERTAINS SECTEURS CENTRES EN 1. CE PHENOMENE EST AUSSI APPELE SURSTABILITE, ET EST COURANT POUR LES EQUATIONS DE CE TYPE, NOTAMMENT (MAIS PAS UNIQUEMENT) EN LIAISON AVEC UNE BIFURCATION DE HOPF. ON ETUDIE TRES PRECISEMENT LE DOMAINE D'EXISTENCE DE CES SOLUTIONS SURSTABLES, PARTICULIEREMENT PRES DE L'AUTRE POINT TOURNANT POUR L'EQUATION, 1, DONT ON REUSSIT A S'APPROCHER, A UNE VITESSE DE L'ORDRE LA RACINE CUBIQUE DU PARAMETRE E. CE RESULTAT, COMBINE A LA DEPENDANCE HOLOMORPHE DES SOLUTIONS EN E, PERMET DE DONNER UN EQUIVALENT DES COEFFICIENTS AN DU DEVELOPPEMENT ASYMPTOTIQUE DES VALEURS DE A EN LE PARAMETRE E (POUR LES A CORRESPONDANT A DES SOLUTIONS SURSTABLES). CE RESULTAT OBTENU POUR L'EQUATION DE VAN DER POL EST ENSUITE GENERALISE A UNE LARGE CLASSE D'EQUATIONS. ON REUSSIT A DEMONTRER L'EXISTENCE DE SOLUTIONS SURSTABLES, ET A DONNER UNE METHODE DE CONSTRUCTION DE LEUR DOMAINE D'EXISTENCE MAXIMAL. ON TRAITE A LA FIN UN DERNIER EXEMPLE, DERIVE DU SYSTEME DU BRUSSELATOR.
Author: ERIC.. MATZINGER
Publisher:
ISBN:
Category :
Languages : fr
Pages : 110
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Book Description
ON PART DE L'EQUATION DE VAN DER POL, E V DV/DU = (1U.U) V + AU, QUI EST CONNUE POUR AVOIR, POUR CERTAINES VALEURS DU PARAMETRE A (VALEUR DEPENDANT DU PARAMETRE DE PERTURBATION E QU'ON FAIT TENDRE VERS 0), DES SOLUTIONS EXCEPTIONNELLES, APPELEES CANARDS, CONTINUES ET BORNEES DANS UN VOISINAGE COMPLET DU POINT 1 ; ALORS QUE CE POINT EST HABITUELLEMENT UN POINT TOURNANT POUR LES SOLUTIONS DE CETTE EQUATION, C'EST-A-DIRE QU'ELLES SONT AU MIEUX BORNEES DANS CERTAINS SECTEURS CENTRES EN 1. CE PHENOMENE EST AUSSI APPELE SURSTABILITE, ET EST COURANT POUR LES EQUATIONS DE CE TYPE, NOTAMMENT (MAIS PAS UNIQUEMENT) EN LIAISON AVEC UNE BIFURCATION DE HOPF. ON ETUDIE TRES PRECISEMENT LE DOMAINE D'EXISTENCE DE CES SOLUTIONS SURSTABLES, PARTICULIEREMENT PRES DE L'AUTRE POINT TOURNANT POUR L'EQUATION, 1, DONT ON REUSSIT A S'APPROCHER, A UNE VITESSE DE L'ORDRE LA RACINE CUBIQUE DU PARAMETRE E. CE RESULTAT, COMBINE A LA DEPENDANCE HOLOMORPHE DES SOLUTIONS EN E, PERMET DE DONNER UN EQUIVALENT DES COEFFICIENTS AN DU DEVELOPPEMENT ASYMPTOTIQUE DES VALEURS DE A EN LE PARAMETRE E (POUR LES A CORRESPONDANT A DES SOLUTIONS SURSTABLES). CE RESULTAT OBTENU POUR L'EQUATION DE VAN DER POL EST ENSUITE GENERALISE A UNE LARGE CLASSE D'EQUATIONS. ON REUSSIT A DEMONTRER L'EXISTENCE DE SOLUTIONS SURSTABLES, ET A DONNER UNE METHODE DE CONSTRUCTION DE LEUR DOMAINE D'EXISTENCE MAXIMAL. ON TRAITE A LA FIN UN DERNIER EXEMPLE, DERIVE DU SYSTEME DU BRUSSELATOR.
Author: Catherine Stenger
Publisher:
ISBN:
Category :
Languages : fr
Pages : 110
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Book Description
Author: Catherine Stenger
Publisher:
ISBN:
Category : EQUATION SYSTEM/DIFFERENTIAL EQUATION/SINGULAR PERTURBATION/EXISTENCE OF SOLUTION/ANALYTICAL SOLUTION/NUMERICAL SOLUTION/ASYMPTOTIC EXPANSION
Languages : fr
Pages : 110
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Book Description
DANS CETTE THESE, NOUS CONSIDERONS DES SYSTEMES D'EQUATIONS DIFFERENTIELLES ORDINAIRES SINGULIEREMENT PERTURBEES PAR UN PETIT PARAMETRE COMPLEXE . S'IL EXISTE UNE SOLUTION FONDAMENTALE FORMELLE DE CE SYSTEME AVEC DES COEFFICIENTS HOLOMORPHES DANS UN VOISINAGE DE X#0, ET SI X#0 EST UN POINT ASYMPTOTIQUEMENT SIMPLE (CE QUI N'EST PAS TOUJOURS LE CAS), ALORS IL EST CONNU QUE, POUR TOUT SECTEUR DU -PLAN D'ANGLE D'OUVERTURE SUFFISAMMENT PETIT, IL EXISTE UNE SOLUTION FONDAMENTALE ADMETTANT COMME DEVELOPPEMENT ASYMPTOTIQUE LA SOLUTION FONDAMENTALE FORMELLE LORSQUE TEND VERS ZERO DANS LE SECTEUR. CECI SIGNIFIE QUE X#0 N'EST PAS UN POINT TOURNANT POUR LE SYSTEME CONSIDERE. LE BUT PRINCIPAL DE CE TRAVAIL EST DE PROUVER LA CONJECTURE DE W. WASOW : X#0 EST UN POINT TOURNANT POUR LE SYSTEME SI ET SEULEMENT SI LA SOLUTION FONDAMENTALE FORMELLE POSSEDE UNE SINGULARITE EN CE POINT. LA DEMARCHE POUR LA DEMONSTRATION EST D'UTILISER LES TECHNIQUES GEVREY POUR PROUVER L'EXISTENCE DE TRANSFORMATIONS QUI DECOMPOSENT LE SYSTEME EN DES SYSTEMES PLUS PETITS QUI SONT ESSENTIELLEMENT SCALAIRES.
Author: Catherine Stenger
Publisher:
ISBN:
Category :
Languages : fr
Pages : 0
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Book Description
Author: Galina Filipuk
Publisher: Springer
ISBN: 3319991485
Category : Mathematics
Languages : en
Pages : 273
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Book Description
These proceedings provide methods, techniques, different mathematical tools and recent results in the study of formal and analytic solutions to Diff. (differential, partial differential, difference, q-difference, q-difference-differential.... ) Equations. They consist of selected contributions from the conference "Formal and Analytic Solutions of Diff. Equations", held at Alcalá de Henares, Spain during September 4-8, 2017. Their topics include summability and asymptotic study of both ordinary and partial differential equations. The volume is divided into four parts. The first paper is a survey of the elements of nonlinear analysis. It describes the algorithms to obtain asymptotic expansion of solutions of nonlinear algebraic, ordinary differential, partial differential equations, and of systems of such equations. Five works on formal and analytic solutions of PDEs are followed by five papers on the study of solutions of ODEs. The proceedings conclude with five works on related topics, generalizations and applications. All contributions have been peer reviewed by anonymous referees chosen among the experts on the subject. The volume will be of interest to graduate students and researchers in theoretical and applied mathematics, physics and engineering seeking an overview of the recent trends in the theory of formal and analytic solutions of functional (differential, partial differential, difference, q-difference, q-difference-differential) equations in the complex domain.
Author: Augustin Fruchard
Publisher: Springer
ISBN: 3642340350
Category : Mathematics
Languages : en
Pages : 169
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Book Description
The purpose of these lecture notes is to develop a theory of asymptotic expansions for functions involving two variables, while at the same time using functions involving one variable and functions of the quotient of these two variables. Such composite asymptotic expansions (CAsEs) are particularly well-suited to describing solutions of singularly perturbed ordinary differential equations near turning points. CAsEs imply inner and outer expansions near turning points. Thus our approach is closely related to the method of matched asymptotic expansions. CAsEs offer two unique advantages, however. First, they provide uniform expansions near a turning point and away from it. Second, a Gevrey version of CAsEs is available and detailed in the lecture notes. Three problems are presented in which CAsEs are useful. The first application concerns canard solutions near a multiple turning point. The second application concerns so-called non-smooth or angular canard solutions. Finally an Ackerberg-O’Malley resonance problem is solved.
Author: Werner Balser
Publisher: Springer Science & Business Media
ISBN: 0387986901
Category : Mathematics
Languages : en
Pages : 314
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Book Description
Simple Ordinary Differential Equations may have solutions in terms of power series whose coefficients grow at such a rate that the series has a radius of convergence equal to zero. In fact, every linear meromorphic system has a formal solution of a certain form, which can be relatively easily computed, but which generally involves such power series diverging everywhere. In this book the author presents the classical theory of meromorphic systems of ODE in the new light shed upon it by the recent achievements in the theory of summability of formal power series.
Author: Werner Balser
Publisher: Springer
ISBN: 3540485945
Category : Mathematics
Languages : en
Pages : 117
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Book Description
Multisummability is a method which, for certain formal power series with radius of convergence equal to zero, produces an analytic function having the formal series as its asymptotic expansion. This book presents the theory of multisummabi- lity, and as an application, contains a proof of the fact that all formal power series solutions of non-linear meromorphic ODE are multisummable. It will be of use to graduate students and researchers in mathematics and theoretical physics, and especially to those who encounter formal power series to (physical) equations with rapidly, but regularly, growing coefficients.
Author: Michèle Loday-Richaud
Publisher: Springer
ISBN: 3319290754
Category : Mathematics
Languages : en
Pages : 286
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Book Description
Addressing the question how to “sum” a power series in one variable when it diverges, that is, how to attach to it analytic functions, the volume gives answers by presenting and comparing the various theories of k-summability and multisummability. These theories apply in particular to all solutions of ordinary differential equations. The volume includes applications, examples and revisits, from a cohomological point of view, the group of tangent-to-identity germs of diffeomorphisms of C studied in volume 1. With a view to applying the theories to solutions of differential equations, a detailed survey of linear ordinary differential equations is provided, which includes Gevrey asymptotic expansions, Newton polygons, index theorems and Sibuya’s proof of the meromorphic classification theorem that characterizes the Stokes phenomenon for linear differential equations. This volume is the second in a series of three, entitled Divergent Series, Summability and Resurgence. It is aimed at graduate students and researchers in mathematics and theoretical physics who are interested in divergent series, Although closely related to the other two volumes, it can be read independently.
