# A Reflection on Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields

01 Jan 2015-

TL;DR: In this paper, the authors introduce differential equations and dynamical systems, including hyperbolic sets, Sympolic Dynamics, and Strange Attractors, and global bifurcations.

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.

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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.

9,201 citations

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TL;DR: A review of the main mathematical ideas and their concrete implementation in analyzing experiments can be found in this paper, where 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).

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.

4,619 citations

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01 Oct 2006

TL;DR: This book explains the relationship of electrophysiology, nonlinear dynamics, and the computational properties of neurons, with each concept presented in terms of both neuroscience and mathematics and illustrated using geometrical intuition, providing a link between the two disciplines.

Abstract: This book explains the relationship of electrophysiology, nonlinear dynamics, and the computational properties of neurons, with each concept presented in terms of both neuroscience and mathematics and illustrated using geometrical intuition In order to model neuronal behavior or to interpret the results of modeling studies, neuroscientists must call upon methods of nonlinear dynamics This book offers an introduction to nonlinear dynamical systems theory for researchers and graduate students in neuroscience It also provides an overview of neuroscience for mathematicians who want to learn the basic facts of electrophysiology "Dynamical Systems in Neuroscience" presents a systematic study of the relationship of electrophysiology, nonlinear dynamics, and computational properties of neurons It emphasizes that information processing in the brain depends not only on the electrophysiological properties of neurons but also on their dynamical properties The book introduces dynamical systems starting with one- and two-dimensional Hodgkin-Huxley-type models and continuing to a description of bursting systems Each chapter proceeds from the simple to the complex, and provides sample problems at the end The book explains all necessary mathematical concepts using geometrical intuition; it includes many figures and few equations, making it especially suitable for non-mathematicians Each concept is presented in terms of both neuroscience and mathematics, providing a link between the two disciplines Nonlinear dynamical systems theory is at the core of computational neuroscience research, but it is not a standard part of the graduate neuroscience curriculum - or taught by math or physics department in a way that is suitable for students of biology This book offers neuroscience students and researchers a comprehensive account of concepts and methods increasingly used in computational neuroscience

3,683 citations

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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.

3,289 citations

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TL;DR: In this article, 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.

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.

3,249 citations

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01 Aug 1983TL;DR: In this article, the authors introduce differential equations and dynamical systems, including hyperbolic sets, Sympolic Dynamics, and Strange Attractors, and global bifurcations.

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.

12,669 citations

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15 Feb 2002

TL;DR: In this paper, the authors present a broad overview of nonlinear phenomena point attractors in autonomous systems, including limit cycles in autonomous system, and chaotic behaviour of one-and two-dimensional maps.

Abstract: Preface. Preface to the First Edition. Acknowledgements from the First Edition. Introduction PART I: BASIC CONCEPTS OF NONLINEAR DYNAMICS An overview of nonlinear phenomena Point attractors in autonomous systems Limit cycles in autonomous systems Periodic attractors in driven oscillators Chaotic attractors in forced oscillators Stability and bifurcations of equilibria and cycles PART II ITERATED MAPS AS DYNAMICAL SYSTEMS Stability and bifurcation of maps Chaotic behaviour of one--and two--dimensional maps PART III FLOWS, OUTSTRUCTURES AND CHAOS The Geometry of Recurrence The Lorenz system Rosslers band Geometry of bifurcations PART IV APPLICATIONS IN THE PHYSICAL SCIENCES Subharmonic resonances of an offshore structure Chaotic motions of an impacting system Escape from a potential well Appendix. Illustrated Glossary. Bibliography. Online Resource. Index.

1,731 citations

01 Jan 1985

TL;DR: Guckenheimer and Holmes as discussed by the authors survey the theory and techniques needed to understand chaotic behavior of ODEs and provide a user's guide to an extensive and rapidly growing field.

Abstract: One important aspect of dynamical systems is the study of the long-term behavior of a set of ordinary differential equations (ODEs) In recent years many systems that are simple to write down have been discovered whose solutions are chaotic They oscillate irregularly, never settling down to a regular pattern Two trajectories which start close together will separate quickly Systems whose time evolution is governed by a parameter p can undergo intriguing variations in the behavior of trajectories In many cases, there are values p* such that the long-term behavior of typical trajectories of p p* For example, the system may go from stable periodic behavior for p p* Such sudden, discontinuous changes or "bifurcations" are quite common Research in chaos and bifurcations in dynamical processes has advanced at a rapid pace during the past decade, acquiring an extraordinary breadth of applications in fields as diverse as fluid mechanics, electrical engineering and neurophysiology The new results interest a wide spectrum of the scientific community, many of whose members, however, lack the mathematical background necessary to decipher the literature Accordingly, Guckenheimer and Holmes have written their book as a "user's guide" to an extensive and rapidly growing field The book surveys the theory and techniques needed to understand chaotic behavior of ODEs The first chapter contains a brief introduction of the theory of ODEs; it is a review of topics usually found in a standard text like Hirsch and Smale (1) The second chapter considers four examples of chaotic systems: the forced van der Pol oscillator, Duffing's equation, the celebrated Lorenz equations, and Holmes' "bouncing ball map" (perhaps more familiar as the map which describes the motion of a periodically forced, damped planar pendulum in the absence of gravity) These examples

1,528 citations

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TL;DR: In this paper, the authors introduce the double-well form of the Duffing equation for nonlinear structural dynamics and compare it to the dynamics of a continuous beam and the behavior of a point mass sliding on a curve.

Abstract: Of the several archetypal equations such as those of Lorenz, Henon, Rossler etc, it is the so-called Duffing equation, and especially its double-well form, that constitutes the backbone of this book. L N Virgin presents basic linear behaviour and useful concepts that are necessary for studying nonlinear behaviour (e.g. Poincare sections, spectral analysis). Then the paradigm of the Duffing equation is introduced within the framework of nonlinear structural dynamics. An analogy is developed between the dynamics of a continuous beam and the behaviour of a point mass sliding on a curve. This latter system is carefully described from a mathematical point of view so that the theoretical bases for construction of experiments and a primary analysis of results are posed. Several experimental models and devices are derived from previous systems (cart movement, Scotch-Yorke forcing, `impact' experiments with cart). Worthwhile numerical values are given that correspond to the dimensions and properties of materials in experimental settings. Theoretical and experimental frameworks are generally clearly explained. The author can provide analytical and/or numerical results together with experimental results. The pleasant presentation of the field of nonlinear oscillations and chaos is continually reinforced by the good agreement that is obtained and shown between numerical and experimental results, even for global dynamics with chaos (e.g. attractors via Poincare maps, basins of attraction). This book will surely be of great interest to students, engineers and researchers involved in the field of nonlinear dynamics. To conclude, let us say that the main disadvantage of this book is that experimental devices are not sold with it! Fortunately, so that one can neglect this `defect', a website address is available: readers can see movies of the most significant experiments that illustrate the theoretical purpose of this most interesting book. C-H Lamarque

109 citations