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Pitchfork bifurcation

About: Pitchfork bifurcation is a research topic. Over the lifetime, 3607 publications have been published within this topic receiving 94797 citations.


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Book
01 Aug 1983
TL;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

Book
01 Jan 1995
TL;DR: One-Parameter Bifurcations of Equilibria in continuous-time systems and fixed points in Discrete-Time Dynamical Systems have been studied in this paper, where they have been used for topological equivalence and structural stability of dynamical systems.
Abstract: Introduction to Dynamical Systems * Topological Equivalence, Bifurcations, and Structural Stability of Dynamical Systems * One-Parameter Bifurcations of Equilibria in Continuous-Time Systems * One-Parameter Bifurcations of Fixed Points in Discrete-Time Systems * Bifurcations of Equilibria and Periodic Orbits in n-Dimensional Systems * Bifurcations of Orbits Homoclinic and Heteroclinic to Hyperbolic Equilibria * Other One-Parameter Bifurcations in Continuous-Time Systems * Two-Parameter Bifurcations of Equilibria in Continuous-Time Dynamical Systems * Two-Parameter Bifurcations of Fixed Points in Discrete-Time Dynamical Systems * Numerical Analysis of Bifurcations * A: Basic Notions from Algebra, Analysis, and Geometry * References * Index.

5,062 citations

Journal ArticleDOI
TL;DR: This book discusses Chaos, Fractals, and Dynamics, and the Importance of Being Nonlinear in a Dynamical View of the World, which aims to clarify the role of Chaos in the world the authors live in.
Abstract: Preface 1. Overview 1.0 Chaos, Fractals, and Dynamics 1.1 Capsule History of Dynamics 1.2 The Importance of Being Nonlinear 1.3 A Dynamical View of the World PART I. ONE-DIMENSIONAL FLOWS 2. Flows on the Line 2.0 Introduction 2.1 A Geometric Way of Thinking 2.2 Fixed Points and Stability 2.3 Population Growth 2.4 Linear Stability Analysis 2.5 Existence and Uniqueness 2.6 Impossibility of Oscillations 2.7 Potentials 2.8 Solving Equations on the Computer Exercises 3. Bifurcations 3.0 Introduction 3.1 Saddle-Node Bifurcation 3.2 Transcritical Bifurcation 3.3 Laser Threshold 3.4 Pitchfork Bifurcation 3.5 Overdamped Bead on a Rotating Hoop 3.6 Imperfect Bifurcations and Catastrophes 3.7 Insect Outbreak Exercises 4. Flows on the Circle 4.0 Introduction 4.1 Examples and Definitions 4.2 Uniform Oscillator 4.3 Nonuniform Oscillator 4.4 Overdamped Pendulum 4.5 Fireflies 4.6 Superconducting Josephson Junctions Exercises PART II. TWO-DIMENSIONAL FLOWS 5. Linear Systems 5.0 Introduction 5.1 Definitions and Examples 5.2 Classification of Linear Systems 5.3 Love Affairs Exercises 6. Phase Plane 6.0 Introduction 6.1 Phase Portraits 6.2 Existence, Uniqueness, and Topological Consequences 6.3 Fixed Points and Linearization 6.4 Rabbits versus Sheep 6.5 Conservative Systems 6.6 Reversible Systems 6.7 Pendulum 6.8 Index Theory Exercises 7. Limit Cycles 7.0 Introduction 7.1 Examples 7.2 Ruling Out Closed Orbits 7.3 Poincare-Bendixson Theorem 7.4 Lienard Systems 7.5 Relaxation Oscillators 7.6 Weakly Nonlinear Oscillators Exercises 8. Bifurcations Revisited 8.0 Introduction 8.1 Saddle-Node, Transcritical, and Pitchfork Bifurcations 8.2 Hopf Bifurcations 8.3 Oscillating Chemical Reactions 8.4 Global Bifurcations of Cycles 8.5 Hysteresis in the Driven Pendulum and Josephson Junction 8.6 Coupled Oscillators and Quasiperiodicity 8.7 Poincare Maps Exercises PART III. CHAOS 9. Lorenz Equations 9.0 Introduction 9.1 A Chaotic Waterwheel 9.2 Simple Properties of the Lorenz Equations 9.3 Chaos on a Strange Attractor 9.4 Lorenz Map 9.5 Exploring Parameter Space 9.6 Using Chaos to Send Secret Messages Exercises 10. One-Dimensional Maps 10.0 Introduction 10.1 Fixed Points and Cobwebs 10.2 Logistic Map: Numerics 10.3 Logistic Map: Analysis 10.4 Periodic Windows 10.5 Liapunov Exponent 10.6 Universality and Experiments 10.7 Renormalization Exercises 11. Fractals 11.0 Introduction 11.1 Countable and Uncountable Sets 11.2 Cantor Set 11.3 Dimension of Self-Similar Fractals 11.4 Box Dimension 11.5 Pointwise and Correlation Dimensions Exercises 12. Strange Attractors 12.0 Introductions 12.1 The Simplest Examples 12.2 Henon Map 12.3 Rossler System 12.4 Chemical Chaos and Attractor Reconstruction 12.5 Forced Double-Well Oscillator Exercises Answers to Selected Exercises References Author Index Subject Index

2,949 citations

Book
01 Jan 1981
TL;DR: The Hopf Bifurcation Theorum has been used in many applications, such as Differential Difference and Integro-differential Equations (by hand).
Abstract: 1. The Hopf Bifurcation Theorum 2. Applications: Ordinary Differential Equations (by hand) 3. Numerical Evaluation of Hopf Bifurcation Formulae 4. Applications: Differential-Difference and Integro-differential Equations (by hand) 5. Applications: Partial Differential Equations (by hand).

2,090 citations

Book
12 Nov 2011
TL;DR: In this paper, the static and dynamic aspects of bifurcation theory, which are of particular pertinence to differential equations, have been discussed, and a discussion of the changes that occur in the structure of the set of zeros of a function as parameters in the function are varied.
Abstract: Having presented background material from functional analysis and the qualitative theory of differential equations, this text focuses on the static and dynamic aspects of bifurcation theory, which are of particular pertinence to differential equations. Static bifurcation theory is concerned with the changes that occur in the structure of the set of zeros of a function as parameters in the function are varied. Dynamic bifurcation theory is concerned with the changes that occur in the structure of the limit sets of solutions of differential equations as parameters in the vector field are varied.

1,848 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202354
2022121
202162
202059
201956
201859