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Van der Pol oscillator

About: Van der Pol oscillator is a research topic. Over the lifetime, 3335 publications have been published within this topic receiving 54257 citations.

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Journal ArticleDOI
TL;DR: Van der Pol's equation for a relaxation oscillator is generalized by the addition of terms to produce a pair of non-linear differential equations with either a stable singular point or a limit cycle, which qualitatively resembles Bonhoeffer's theoretical model for the iron wire model of nerve.

5,430 citations

06 Jan 1994
TL;DR: The phenomenology of chaos Towards a theory of nonlinear dynamics and chaos Quantifying chaos Special topics Appendices Index
Abstract: First Edition Preface First Edition Acknowledgments Second Edition Preface Second Edition Acknowledgments I. THE PHENOMENOLOGY OF CHAOS 1. Three Chaotic Systems 2. The Universality of Chaos II. TOWARDS A THEORY OF NONLINEAR DYNAMICS AND CHAOS 3. Dynamics in State Space: One and Two Dimensions 4. Three-Dimensional State Space and Chaos 5. Iterated Maps 6. Quasi-Periodicity and Chaos 7. Intermittency and Crises 8. Hamiltonian Systems III.MEASURES OF CHAOS 9. Quantifying Chaos 10. Many Dimensions and Multifractals IV.SPECIAL TOPICS 11. Pattern Formation and Spatiotemporal Chaos 12. Quantum Chaos, The Theory of Complexity, and other Topics Appendix A: Fourier Power Spectra Appendix B: Bifurcation Theory Appendix C: The Lorenz Model Appendix D: The Research Literature on Chaos Appendix E: Computer Programs Appendix F: Theory of the Universal Feigenbaum Numbers Appendix G: The Duffing Double-Well Oscillator Appendix H: Other Universal Feature for One-Dimensional Iterated Maps Appendix I: The van der Pol Oscillator Appendix J: Simple Laser Dynamics Models References Bibliography Index

1,055 citations

01 Jan 1995
TL;DR: In this paper, the authors present a mathematical model for time-series analysis of human heart rate response to Sinusoid inputs, showing that it is a function of the number of neurons in the human heart.
Abstract: 1 Finite-Difference Equations.- 1.1 A Mythical Field.- 1.2 The Linear Finite-Difference Equation.- 1.3 Methods of Iteration.- 1.4 Nonlinear Finite-Difference Equations.- 1.5 Steady States and Their Stability.- 1.6 Cycles and Their Stability.- 1.7 Chaos.- 1.8 Quasiperiodicity.- 1 Chaos in Periodically Stimulated Heart Cells.- Sources and Notes.- Exercises.- Computer Projects.- 2 Boolean Networks and Cellular Automata.- 2.1 Elements and Networks.- 2.2 Boolean Variables, Functions, and Networks.- 2 A Lambda Bacteriophage Model.- 3 Locomotion in Salamanders.- 2.3 Boolean Functions and Biochemistry.- 2.4 Random Boolean Networks.- 2.5 Cellular Automata.- 4 Spiral Waves in Chemistry and Biology.- 2.6 Advanced Topic: Evolution and Computation.- Sources and Notes.- Exercises.- Computer Projects.- 3 Self-Similarity and Fractal Geometry.- 3.1 Describing a Tree.- 3.2 Fractals.- 3.3 Dimension.- 5 The Box-Counting Dimension.- 3.4 Statistical Self-Similarity.- 6 Self-Similarity in Time.- 3.5 Fractals and Dynamics.- 7 Random Walks and Levy Walks.- 8 Fractal Growth.- Sources and Notes.- Exercises.- Computer Projects.- 4 One-Dimensional Differential Equations.- 4.1 Basic Definitions.- 4.2 Growth and Decay.- 9 Traffic on the Internet.- 10 Open Time Histograms in Patch Clamp Experiments.- 11 Gompertz Growth of Tumors.- 4.3 Multiple Fixed Points.- 4.4 Geometrical Analysis of One-Dimensional Nonlinear Ordinary Differential Equations.- 4.5 Algebraic Analysis of Fixed Points.- 4.6 Differential Equations versus Finite-Difference Equations.- 4.7 Differential Equations with Inputs.- 12 Heart Rate Response to Sinusoid Inputs.- 4.8 Advanced Topic: Time Delays and Chaos.- 13 Nicholson's Blowflies.- Sources and Notes.- Exercises.- Computer Projects.- 5 Two-Dimensional Differential Equations.- 5.1 The Harmonic Oscillator.- 5.2 Solutions, Trajectories, and Flows.- 5.3 The Two-Dimensional Linear Ordinary Differential Equation.- 5.4 Coupled First-Order Linear Equations.- 14 Metastasis of Malignant Tumors.- 5.5 The Phase Plane.- 5.6 Local Stability Analysis of Two-Dimensional, Nonlinear Differential Equations.- 5.7 Limit Cycles and the van der Pol Oscillator.- 5.8 Finding Solutions to Nonlinear Differential Equations.- 15 Action Potentials in Nerve Cells.- 5.9 Advanced Topic: Dynamics in Three or More Dimensions.- 5.10 Advanced Topic: Poincare Index Theorem.- Sources and Notes.- Exercises.- Computer Projects.- 6 Time-Series Analysis.- 6.1 Starting with Data.- 6.2 Dynamics, Measurements, and Noise.- 16 Fluctuations in Marine Populations.- 6.3 The Mean and Standard Deviation.- 6.4 Linear Correlations.- 6.5 Power Spectrum Analysis.- 17 Daily Oscillations in Zooplankton.- 6.6 Nonlinear Dynamics and Data Analysis.- 18 Reconstructing Nerve Cell Dynamics.- 6.7 Characterizing Chaos.- 19 Predicting the Next Ice Age.- 6.8 Detecting Chaos and Nonlinearity.- 6.9 Algorithms and Answers.- Sources and Notes.- Exercises.- Computer Projects.- Appendix A A Multi-Functional Appendix.- A.1 The Straight Line.- A.2 The Quadratic Function.- A.3 The Cubic and Higher-Order Polynomials.- A.4 The Exponential Function.- A.5 Sigmoidal Functions.- A.6 The Sine and Cosine Functions.- A.7 The Gaussian (or "Normal") Distribution.- A.8 The Ellipse.- A.9 The Hyperbola.- Exercises.- Appendix B A Note on Computer Notation.- Solutions to Selected Exercises.

759 citations

Journal ArticleDOI
TL;DR: A class of low-order models for vortex-induced vibrations is analyzed in this article, where a van der Pol equation is used to describe the near wake dynamics describing the fluctuating nature of vortex shedding and several types of linear coupling terms modelling the fluid-structure interaction are considered.

616 citations

Journal ArticleDOI
TL;DR: In this paper, a geometric analysis of relaxation oscillations and canard cycles in singularly perturbed planar vector fields is presented. But the analysis is restricted to the case of singular cycles and does not consider the case where the transition from small Hopf-type cycles to large relaxation cycles occurs in an exponentially thin parameter interval.

465 citations

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