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Journal ArticleDOI

Problème général de la stabilité du mouvement

01 Jan 1907-Annales de la Faculté des Sciences de Toulouse (GAUTHIER-VILLARS, IMPRIMEUR-EDITEUR ; ED. PRIVAT, IMPRIMEUR-LIBRAIRE)-Vol. 9, pp 203-474
TL;DR: In this article, Sabatier implique l'accord avec les conditions générales d'utilisation (http://www.up-tlse.numdam.org/conditions).
Abstract: © Université Paul Sabatier, 1907, tous droits réservés. L’accès aux archives de la revue « Annales de la faculté des sciences de Toulouse » (http://picard.ups-tlse.fr/~annales/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

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Book
19 Aug 1998
TL;DR: This chapter establishes the framework of random dynamical systems and introduces the concept of random attractors to analyze models with stochasticity or randomness.
Abstract: I. Random Dynamical Systems and Their Generators.- 1. Basic Definitions. Invariant Measures.- 2. Generation.- II. Multiplicative Ergodic Theory.- 3. The Multiplicative Ergodic Theorem in Euclidean Space.- 4. The Multiplicative Ergodic Theorem on Bundles and Manifolds.- 5. The MET for Related Linear and Affine RDS.- 6. RDS on Homogeneous Spaces of the General Linear Group.- III. Smooth Random Dynamical Systems.- 7. Invariant Manifolds.- 8. Normal Forms.- 9. Bifurcation Theory.- IV. Appendices.- Appendix A. Measurable Dynamical Systems.- A.1 Ergodic Theory.- A.2 Stochastic Processes and Dynamical Systems.- A.3 Stationary Processes.- A.4 Markov Processes.- Appendix B. Smooth Dynamical Systems.- B.1 Two-Parameter Flows on a Manifold.- B.4 Autonomous Case: Dynamical Systems.- B.5 Vector Fields and Flows on Manifolds.- References.

2,663 citations

Book
15 Jun 2001
TL;DR: The Time Scales Calculus as discussed by the authors is a generalization of the time-scales calculus with linear systems and higher-order linear equations, and it can be expressed in terms of linear Symplectic Dynamic Systems.
Abstract: Preface * The Time Scales Calculus * First Order Linear Equations * Second Order Linear Equations * Self-Adjoint Equations * Linear Systems and Higher Order Equations * Dynamic Inequalities * Linear Symplectic Dynamic Systems * Extensions * Solutions to Selected Problems * Bibliography * Index

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Book
30 Jul 1997
TL;DR: This paper presents a meta-modelling procedure called Multimachine Dynamic Models for Energy Function Methods, which automates the very labor-intensive and therefore time-heavy and expensive process of Synchronous Machine Modeling.
Abstract: 1 Introduction 2 Electromagnetic Transients 3 Synchronous Machine Modeling 4 Synchronous Machine Control Models 5 Single-Machine Dynamic Models 6 Multimachine Dynamic Models 7 Multimachine Simulation 8 Small-Signal Stability 9 Energy Function Methods Appendix A: Integral Manifolds for Model Bibliography Index

2,004 citations

Journal ArticleDOI
TL;DR: In this paper, a method for computing all of the Lyapunov characteristic exponents of order greater than one is presented, which is related to the increase of volumes of a dynamical system.
Abstract: Since several years Lyapunov Characteristic Exponents are of interest in the study of dynamical systems in order to characterize quantitatively their stochasticity properties, related essentially to the exponential divergence of nearby orbits. One has thus the problem of the explicit computation of such exponents, which has been solved only for the maximal of them. Here we give a method for computing all of them, based on the computation of the exponents of order greater than one, which are related to the increase of volumes. To this end a theorem is given relating the exponents of order one to those of greater order. The numerical method and some applications will be given in a forthcoming paper.

1,659 citations

Journal ArticleDOI
TL;DR: A stability theorem for systems described by IQCs is presented that covers classical passivity/dissipativity arguments but simplifies the use of multipliers and the treatment of causality.
Abstract: This paper introduces a unified approach to robustness analysis with respect to nonlinearities, time variations, and uncertain parameters. From an original idea by Yakubovich (1967), the approach has been developed under a combination of influences from the Western and Russian traditions of control theory. It is shown how a complex system can be described, using integral quadratic constraints (IQC) for its elementary components. A stability theorem for systems described by IQCs is presented that covers classical passivity/dissipativity arguments but simplifies the use of multipliers and the treatment of causality. A systematic computational approach is described, and relations to other methods of stability analysis are discussed. Last, but not least, the paper contains a summarizing list of IQCs for important types of system components.

1,547 citations

References
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Journal ArticleDOI
TL;DR: In this paper, Gauthier-Villars implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions).
Abstract: © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1883, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

1,242 citations