Topic

# Hopf bifurcation

About: Hopf bifurcation is a research topic. Over the lifetime, 9726 publications have been published within this topic receiving 181965 citations. The topic is also known as: Poincaré–Andronov–Hopf bifurcation & Andronov–Hopf bifurcation.

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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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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

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17 Aug 1976TL;DR: The Hopf bifurcation refers to the development of periodic orbits ("self-oscillations") from a stable fixed point, as a parameter crosses a critical value as mentioned in this paper.

Abstract: The goal of these notes is to give a reasonably complete, although not exhaustive, discussion of what is commonly referred to as the Hopf bifurcation with applications to specific problems, including stability calculations Historically, the subject had its origins in the works of Poincare [1] around 1892 and was extensively discussed by Andronov and Witt [1] and their co-workers starting around 1930 Hopf's basic paper [1] appeared in 1942 Although the term "Poincare-Andronov-Hopf bifurcation" is more accurate (sometimes Friedrichs is also included), the name "Hopf Bifurcation" seems more common, so we have used it Hopf's crucial contribution was the extension from two dimensions to higher dimensions
The principal technique employed in the body of the text is that of invariant manifolds The method of Ruelle-Takens [1] is followed, with details, examples and proofs added Several parts of the exposition in the main text come from papers of P Chernoff, J Dorroh, O Lanford and F Weissler to whom we are grateful
The general method of invariant manifolds is common in dynamical systems and in ordinary differential equations; see for example, Hale [1,2] and Hartman [1] Of course, other methods are also available In an attempt to keep the picture balanced, we have included samples of alternative approaches Specifically, we have included a translation (by L Howard and N Kopell) of Hopf's original (and generally unavailable) paper These original methods, using power series and scaling are used in fluid mechanics by, amongst many others, Joseph and Sattinger [1]; two sections on these ideas from papers of Iooss [1-6] and Kirchgassner and Kielhoffer [1] (contributed by G Childs and O Ruiz) are given
The contributions of S Smale, J Guckenheimer and G Oster indicate applications to the biological sciences and that of D Schmidt to Hamiltonian systems For other applications and related topics, we refer to the monographs of Andronov and Chaiken [1], Minorsky [1] and Thom [1]
The Hopf bifurcation refers to the development of periodic orbits ("self-oscillations") from a stable fixed point, as a parameter crosses a critical value In Hopf's original approach, the determination of the stability of the resulting periodic orbits is, in concrete problems, an unpleasant calculation We have given explicit algorithms for this calculation which are easy to apply in examples (See Section 4, and Section 5A for comparison with Hopf's formulae) The method of averaging, exposed here by S Chow and J Mallet-Paret in Section 4C gives another method of determining this stability, and seems to be especially useful for the next bifurcation to invariant tori where the only recourse may be to numerical methods, since the periodic orbit is not normally known explicitly
In applications to partial differential equations, the key assumption is that the semi-flow defined by the equations be smooth in all variables for t > O This enables the invariant manifold machinery, and hence the bifurcation theorems to go through (Marsden [2]) To aid in determining smoothness in examples we have presented parts of the results of Dorroh-Marsden [1] Similar ideas for utilizing smoothness have been introduced independently by other authors, such as D Henry [1]
Some further directions of research and generalization are given in papers of Jost and Zehnder [1], Takens [1, 2], Crandall-Rabinowitz [1, 2], Arnold [2], and Kopell-Howard [1-6] to mention just a few that are noted but are not discussed in any detail here We have selected results of Chafee [1] and Ruelle [3] (the latter is exposed here by S Schecter) to indicate some generalizations that are possible
The subject is by no means closed Applications to instabilities in biology (see, eg Zeeman [2], Gurel [1-12] and Section 10, 11); engineering (for example, spontaneous "flutter" or oscillations in structural, electrical, nuclear or other engineering systems; cf Aronson [1], Ziegler [1] and Knops and Wilkes [1]), and oscillations in the atmosphere and the earth's magnetic field (cf Durand [1]) are appearing at a rapid rate Also, the qualitative theory proposed by Ruelle-Takens [1] to describe turbulence is not yet well understood (see Section 9) In this direction, the papers of Newhouse and Peixoto [1] and Alexander and Yorke [1] seem to be important Stable oscillations in nonlinear waves may be another fruitful area for application; cf Whitham [1] We hope these notes provide some guidance to the field and will be useful to those who wish to study or apply these fascinating methods
After we completed our stability calculations we were happy to learn that others had found similar difficulty in applying Hopf's result as it had existed in the literature to concrete examples in dimension ≥ 3 They have developed similar formulae to deal with the problem; cf Hsu and Kazarinoff [1, 2] and Poore [1]
The other main new result here is our proof of the validity of the Hopf bifurcation theory for nonlinear partial differential equations of parabolic type The new proof, relying on invariant manifold theory, is considerably simpler than existing proofs and should be useful in a variety of situations involving bifurcation theory for evolution equations
These notes originated in a seminar given at Berkeley in 1973-4 We wish to thank those who contributed to this volume and wish to apologize in advance for the many important contributions to the field which are not discussed here; those we are aware of are listed in the bibliography which is, admittedly, not exhaustive Many other references are contained in the lengthy bibliography in Cesari [1] We also thank those who have taken an interest in the notes and have contributed valuable comments These include R Abraham, D Aronson, A Chorin, M Crandall, R Cushman, C Desoer, A Fischer, L Glass, J M Greenberg, O Gurel, J Hale, B Hassard, S Hastings, M Hirsch, E Hopf, N D Kazarinoff, J P LaSalle, A Mees, C Pugh, D Ruelle, F Takens, Y Wan and A Weinstein Special thanks go to J A Yorke for informing us of the material in Section 3C and to both he and D Ruelle for pointing out the example of the Lorentz equations (See Example 4B8) Finally, we thank Barbara Komatsu and Jody Anderson for the beautiful job they did in typing the manuscript
Jerrold Marsden
Marjorie McCracken

1,878 citations

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25 Oct 1985

TL;DR: The history of the theory of averaging can be found in this paper, where a 4-dimensional example of Hopf Bifurcation is presented. But the history of averaging is not complete.

Abstract: Basic Material and Asymptotics.- Averaging: the Periodic Case.- Methodology of Averaging.- Averaging: the General Case.- Attraction.- Periodic Averaging and Hyperbolicity.- Averaging over Angles.- Passage Through Resonance.- From Averaging to Normal Forms.- Hamiltonian Normal Form Theory.- Classical (First-Level) Normal Form Theory.- Nilpotent (Classical) Normal Form.- Higher-Level Normal Form Theory.- The History of the Theory of Averaging.- A 4-Dimensional Example of Hopf Bifurcation.- Invariant Manifolds by Averaging.- Some Elementary Exercises in Celestial Mechanics.- On Averaging Methods for Partial Differential Equations.

1,765 citations

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26 Sep 1996

TL;DR: In this paper, the existence and compactness of solution semiflows of linear systems are investigated. But the authors focus on the nonhomogeneous systems and do not consider the linearized stability of non-homogeneous solutions.

Abstract: 1. Preliminaries.- 1.1 Semigroups and generators.- 1.2 Function spaces, elliptic operators, and maximal principles.- Bibliographical Notes.- 2. Existence and Compactness of Solution Semiflows.- 2.1 Existence and compactness.- 2.2 Local existence and global continuation.- 2.3 Extensions to neutral partial functional differential equations.- Bibliographical Notes.- 3. Generators and Decomposition of State Spaces for Linear Systems.- 3.1 Infinitesimal generators of solution semiflows of linear systems.- 3.2 Decomposition of state spaces by invariant subspaces.- 3.3 Computation of center, stable, and unstable subspaces.- 3.4 Extensions to equations with infinite delay.- 3.5 L2-stability and reduction of neutral equations.- Bibliographical Notes.- 4. Nonhomogeneous Systems and Linearized Stability.- 4.1 Dual operators and an alternative theorem.- 4.2 Variation of constants formula.- 4.3 Existence of periodic or almost periodic solutions.- 4.4 Principle of linearized stability.- 4.5 Fundamental transformations and representations of solutions.- Bibliographical Notes.- 5. Invariant Manifolds of Nonlinear Systems.- 5.1 Stable and unstable manifolds.- 5.2 Center manifolds.- 5.3 Flows on center manifolds.- 5.4 Global invariant manifolds of perturbed wave equations.- Bibliographical Notes.- 6. Hopf Bifurcations.- 6.1 Some classical Hopf bifurcation theorems for ODEs.- 6.2 Smooth local Hopf bifurcations: a special case.- 6.3 Some examples from population dynamics.- 6.4 Smooth local Hopf bifurcations: general situations.- 6.5 A topological global Hopf bifurcation theory.- 6.6 Global continuation of wave solutions.- Bibliographical Notes.- 7. Small and Large Diffusivity.- 7.1 Destablization of periodic solutions by small diffusivity.- 7.2 Large diffusivity under Neumann boundary conditions.- Bibliographical Notes.- 8. Invariance, Comparison, and Upper and Lower Solutions.- 8.1 Invariance and inequalities.- 8.2 Systems and strict inequalities.- 8.3 Applications to reaction diffusion equations with delay.- Bibliographical Notes.- 9. Convergence, Monotonicity, and Contracting Rectangles.- 9.1 Monotonicity and generic convergence.- 9.2 Stability and steady state solutions of quasimonotone systems.- 9.3 Comparison and convergence results.- 9.4 Applications to Lotka-Volterra competition models.- Bibliographical Notes.- 10. Dispativeness, Exponential Growth, and Invariance Principles.- 10.1 Point dispativeness in a scalar equation.- 10.2 Convergence in a scalar equation.- 10.3 Exponential growth in a scalar equation.- 10.4 An invariance principle.- Bibliographical Notes.- 11. Traveling Wave Solutions.- 11.1 Huxley nonlinearities and phase plane arguments.- 11.2 Delayed Fisher equation: sub-super solution method.- 11.3 Systems and monotone iteration method.- 11.4 Traveling oscillatory waves.- Bibliographical Notes.

1,747 citations