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Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields

01 Aug 1983-

Abstract: Contents: Introduction: Differential Equations and Dynamical Systems.- An Introduction to Chaos: Four Examples.- Local Bifurcations.- Averaging and Perturbation from a Geometric Viewpoint.- Hyperbolic Sets, Sympolic Dynamics, and Strange Attractors.- Global Bifurcations.- Local Codimension Two Bifurcations of Flows.- Appendix: Suggestions for Further Reading. Postscript Added at Second Printing. Glossary. References. Index.
Topics: Pitchfork bifurcation (59%), Heteroclinic bifurcation (58%), Dynamical systems theory (57%), Bifurcation theory (55%), Attractor (55%)
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01 Jan 2015-
Abstract: Contents: Introduction: Differential Equations and Dynamical Systems.- An Introduction to Chaos: Four Examples.- Local Bifurcations.- Averaging and Perturbation from a Geometric Viewpoint.- Hyperbolic Sets, Sympolic Dynamics, and Strange Attractors.- Global Bifurcations.- Local Codimension Two Bifurcations of Flows.- Appendix: Suggestions for Further Reading. Postscript Added at Second Printing. Glossary. References. Index.

12,485 citations


Journal ArticleDOI
Louis M. Pecora1, Thomas L. Carroll1Institutions (1)
TL;DR: This chapter describes the linking of two chaotic systems with a common signal or signals and highlights that when the signs of the Lyapunov exponents for the subsystems are all negative the systems are synchronized.
Abstract: Certain subsystems of nonlinear, chaotic systems can be made to synchronize by linking them with common signals. The criterion for this is the sign of the sub-Lyapunov exponents. We apply these ideas to a real set of synchronizing chaotic circuits.

8,783 citations


Journal ArticleDOI
Jean-Pierre Eckmann1, David Ruelle2Institutions (2)
Abstract: Physical and numerical experiments show that deterministic noise, or chaos, is ubiquitous. While a good understanding of the onset of chaos has been achieved, using as a mathematical tool the geometric theory of differentiable dynamical systems, moderately excited chaotic systems require new tools, which are provided by the ergodic theory of dynamical systems. This theory has reached a stage where fruitful contact and exchange with physical experiments has become widespread. The present review is an account of the main mathematical ideas and their concrete implementation in analyzing experiments. The main subjects are the theory of dimensions (number of excited degrees of freedom), entropy (production of information), and characteristic exponents (describing sensitivity to initial conditions). The relations between these quantities, as well as their experimental determination, are discussed. The systematic investigation of these quantities provides us for the first time with a reasonable understanding of dynamical systems, excited well beyond the quasiperiodic regimes. This is another step towards understanding highly turbulent fluids.

4,512 citations


Journal ArticleDOI
Michael S. Branicky1Institutions (1)
TL;DR: Bendixson's theorem is extended to the case of Lipschitz continuous vector fields, allowing limit cycle analysis of a class of "continuous switched" systems.
Abstract: We introduce some analysis tools for switched and hybrid systems. We first present work on stability analysis. We introduce multiple Lyapunov functions as a tool for analyzing Lyapunov stability and use iterated function systems theory as a tool for Lagrange stability. We also discuss the case where the switched systems are indexed by an arbitrary compact set. Finally, we extend Bendixson's theorem to the case of Lipschitz continuous vector fields, allowing limit cycle analysis of a class of "continuous switched" systems.

3,136 citations


Cites background from "Nonlinear Oscillations, Dynamical S..."

  • ...Bendixson’s theorem, as it is traditionally stated (e.g., [ 29 ] and [30]), requires continuously differentiable vector fields and is thus not of use in general....

    [...]


Journal ArticleDOI
Abstract: The problem of defining and classifying power system stability has been addressed by several previous CIGRE and IEEE Task Force reports. These earlier efforts, however, do not completely reflect current industry needs, experiences and understanding. In particular, the definitions are not precise and the classifications do not encompass all practical instability scenarios. This report developed by a Task Force, set up jointly by the CIGRE Study Committee 38 and the IEEE Power System Dynamic Performance Committee, addresses the issue of stability definition and classification in power systems from a fundamental viewpoint and closely examines the practical ramifications. The report aims to define power system stability more precisely, provide a systematic basis for its classification, and discuss linkages to related issues such as power system reliability and security.

2,893 citations


Cites background from "Nonlinear Oscillations, Dynamical S..."

  • ...a major advantage of the Lyapunov approach; it also connects naturally with studies of bifurcations [64] that have been of great interest in power systems, mostly related to the topic of voltage collapse....

    [...]


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No. of citations received by the Paper in previous years
YearCitations
202214
2021353
2020333
2019291
2018262
2017348