Theory and Applications of Hopf Bifurcation

Theory and Applications of Hopf Bifurcation

This text will be of value to all those interested in and studying the subject in the mathematical, natural and engineering sciences.

Author: B. D. Hassard

Publisher: CUP Archive

ISBN: 0521231582

Category: Mathematics

Page: 324

View: 587

This text will be of value to all those interested in and studying the subject in the mathematical, natural and engineering sciences.
Categories: Mathematics

The Hopf Bifurcation and Its Applications

The Hopf Bifurcation and Its Applications

SECTION 3C HOPF'S BIFURCATION THEOREM AND THE CENTER THEOREM OF LIAPUNOV by Dieter S. Schmidt Introduction In recent years numerous papers have dealt with the bifurcation of periodic orbits from an equilibrium point.

Author: J. E. Marsden

Publisher: Springer Science & Business Media

ISBN: 9781461263746

Category: Mathematics

Page: 408

View: 595

The goal of these notes is to give a reasonahly com plete, although not exhaustive, discussion of what is commonly referred to as the Hopf bifurcation with applications to spe cific problems, including stability calculations. Historical ly, 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.
Categories: Mathematics

Frequency domain Approach To Hopf Bifurcation Analysis Continuous Time delayed Systems

Frequency domain Approach To Hopf Bifurcation Analysis  Continuous Time delayed Systems

Malek-Zavarei, M. and Jamshidi, M. (1987) Time-delay Systems - Analysis, Optimization and Applications, North-Holland ... Marsden, J. E. and McCracken, M. (1976) The Hopf Bifurcation and Its Applications, Springer-Verlag, New York.

Author: Franco Sebastian Gentile

Publisher: World Scientific

ISBN: 9789811205484

Category: Technology & Engineering

Page: 392

View: 499

This book is devoted to the study of an effective frequency-domain approach, based on systems control theory, to compute and analyze several types of standard bifurcation conditions for general continuous-time nonlinear dynamical systems. A very rich pictorial gallery of local bifurcation diagrams for such nonlinear systems under simultaneous variations of several system parameters is presented. Some higher-order harmonic balance approximation formulas are derived for analyzing the oscillatory dynamics in small neighborhoods of certain types of Hopf and degenerate Hopf bifurcations.The frequency-domain approach is then extended to the large class of delay-differential equations, where the time delays can be either discrete or distributed. For the case of discrete delays, two alternatives are presented, depending on the structure of the underlying dynamical system, where the more general setting is then extended to the case of distributed time-delayed systems. Some representative examples in engineering and biology are discussed.
Categories: Technology & Engineering

Hopf Bifurcation Analysis

Hopf Bifurcation Analysis

This book is devoted to the frequency domain approach, for both regular and degenerate Hopf bifurcation analyses.

Author: Jorge L. Moiola

Publisher: World Scientific

ISBN: 9810226284

Category: Mathematics

Page: 354

View: 552

This book is devoted to the frequency domain approach, for both regular and degenerate Hopf bifurcation analyses. Besides showing that the time and frequency domain approaches are in fact equivalent, the fact that many significant results and computational formulas obtained in the studies of regular and degenerate Hopf bifurcations from the time domain approach can be translated and reformulated into the corresponding frequency domain setting, and be reconfirmed and rediscovered by using the frequency domain methods, is also explained. The description of how the frequency domain approach can be used to obtain several types of standard bifurcation conditions for general nonlinear dynamical systems is given as well as is demonstrated a very rich pictorial gallery of local bifurcation diagrams for nonlinear systems under simultaneous variations of several system parameters. In conjunction with this graphical analysis of local bifurcation diagrams, the defining and nondegeneracy conditions for several degenerate Hopf bifurcations is presented. With a great deal of algebraic computation, some higher-order harmonic balance approximation formulas are derived, for analyzing the dynamical behavior in small neighborhoods of certain types of degenerate Hopf bifurcations that involve multiple limit cycles and multiple limit points of periodic solutions. In addition, applications in chemical, mechanical and electrical engineering as well as in biology are discussed. This book is designed and written in a style of research monographs rather than classroom textbooks, so that the most recent contributions to the field can be included with references.
Categories: Mathematics

Hopf Bifurcation Analysis

Hopf Bifurcation Analysis

Marsden, J. E. & McCracken, M. (1976] The Hopf Bifurcation and Its Applications, Springer-Verlag, New York. Mees, A. I. [1981) Dynamics of Feedback Systems, John Wiley & Sons, Chichester, UK. Mees, A. I. & Allwright, D. J. (1979) “Using ...

Author: Jorge L Moiola

Publisher: World Scientific

ISBN: 9789814499101

Category: Science

Page: 344

View: 749

This book is devoted to the frequency domain approach, for both regular and degenerate Hopf bifurcation analyses. Besides showing that the time and frequency domain approaches are in fact equivalent, the fact that many significant results and computational formulas obtained in the studies of regular and degenerate Hopf bifurcations from the time domain approach can be translated and reformulated into the corresponding frequency domain setting, and be reconfirmed and rediscovered by using the frequency domain methods, is also explained. The description of how the frequency domain approach can be used to obtain several types of standard bifurcation conditions for general nonlinear dynamical systems is given as well as is demonstrated a very rich pictorial gallery of local bifurcation diagrams for nonlinear systems under simultaneous variations of several system parameters. In conjunction with this graphical analysis of local bifurcation diagrams, the defining and nondegeneracy conditions for several degenerate Hopf bifurcations is presented. With a great deal of algebraic computation, some higher-order harmonic balance approximation formulas are derived, for analyzing the dynamical behavior in small neighborhoods of certain types of degenerate Hopf bifurcations that involve multiple limit cycles and multiple limit points of periodic solutions. In addition, applications in chemical, mechanical and electrical engineering as well as in biology are discussed. This book is designed and written in a style of research monographs rather than classroom textbooks, so that the most recent contributions to the field can be included with references. Contents: IntroductionThe Hopf Bifurcation TheoremContinuation of Bifurcation Curves on the Parameter PlaneDegenerate Bifurcations in the Space of System ParametersHigh-Order Hopf Bifurcation FormulasHopf Bifurcation in Nonlinear Systems with Time DelaysBirth of Multiple Limit CyclesAppendixReferencesArthur IndexSubject Index Readership: Nonlinear scientists, applied mathematicians, and systems engineers. keywords:Bifurcation;Harmonic Balance Approximation;Graphical Hopf Bifurcation;Degenerate Hopf Bifurcation;High-Order Hopf Bifurcation;Multiple Limit Cycles;Hopf;Frequency;Harmonic Balance;Feedback;Oscillations;Nonlinear;Delay;Limit Cycles;Degenerate Bifurcations
Categories: Science

Continuation and Bifurcations Numerical Techniques and Applications

Continuation and Bifurcations  Numerical Techniques and Applications

Marsden J.E. and McCracken M. (1976) The Hopf Bifurcation and its Applications, Springer, New York. Meyer K. R. and Schmidt D. S. (1977) Entrainment Domains, Funkcialaj Ekvacioj 20, 171–192. Rand R. H. (1985) Derivation of the Hopf ...

Author: Dirk Roose

Publisher: Springer Science & Business Media

ISBN: 9789400906594

Category: Mathematics

Page: 444

View: 752

Proceedings of the NATO Advanced Research Workshop, Leuven, Belgium, September 18-22, 1989
Categories: Mathematics

Bifurcation Control

Bifurcation Control

The book is not only aimed at active researchers in the field of bifurcation control and its applications, but also at a general audience in related fields.

Author: Guanrong Chen

Publisher: Springer Science & Business Media

ISBN: 3540403418

Category: Technology & Engineering

Page: 344

View: 563

Bifurcation control refers to the task of designing a controller that can modify the bifurcation properties of a given nonlinear system, so as to achieve some desirable dynamical behaviors. There exists no similar control theory-oriented book available in the market that is devoted to the subject of bifurcation control, written by control engineers for control engineers. World-renowned leading experts in the field provide their state-of-the-art survey about the extensive research that has been done over the last few years in this subject. The book is not only aimed at active researchers in the field of bifurcation control and its applications, but also at a general audience in related fields.
Categories: Technology & Engineering

Mathematical Modeling and Applications in Nonlinear Dynamics

Mathematical Modeling and Applications in Nonlinear Dynamics

Karaoglu, E., Merdan, H.: Hopf bifurcation analysis for a ratio-dependent predator-prey system involving two delays. ... J.E., McCracken, M.: The Hopf Bifurcation and Its Applications. Springer, New York (1976) 29.

Author: Albert C.J. Luo

Publisher: Springer

ISBN: 9783319266305

Category: Technology & Engineering

Page: 205

View: 125

The book covers nonlinear physical problems and mathematical modeling, including molecular biology, genetics, neurosciences, artificial intelligence with classical problems in mechanics and astronomy and physics. The chapters present nonlinear mathematical modeling in life science and physics through nonlinear differential equations, nonlinear discrete equations and hybrid equations. Such modeling can be effectively applied to the wide spectrum of nonlinear physical problems, including the KAM (Kolmogorov-Arnold-Moser (KAM)) theory, singular differential equations, impulsive dichotomous linear systems, analytical bifurcation trees of periodic motions, and almost or pseudo- almost periodic solutions in nonlinear dynamical systems.
Categories: Technology & Engineering

Computational Science and Its Applications ICCSA 2016

Computational Science and Its Applications   ICCSA 2016

In: Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 451–455 (2006) 4. Elaydi, S.: Discrete Chaos: ... 44, 267–284 (1992) Marsden, J., McCracken, M.: The Hopf Bifurcation and Its Application.

Author: Osvaldo Gervasi

Publisher: Springer

ISBN: 9783319421117

Category: Computers

Page: 574

View: 234

The five-volume set LNCS 9786-9790 constitutes the refereed proceedings of the 16th International Conference on Computational Science and Its Applications, ICCSA 2016, held in Beijing, China, in July 2016. The 239 revised full papers and 14 short papers presented at 33 workshops were carefully reviewed and selected from 849 submissions. They are organized in five thematical tracks: computational methods, algorithms and scientific applications; high performance computing and networks; geometric modeling, graphics and visualization; advanced and emerging applications; and information systems and technologies.
Categories: Computers

Bifurcation and Chaos

Bifurcation and Chaos

Paris 190, 256–258 (1930) B.D. Hassard, N.D. Kazarinoff and Y. H. Wan: Theory and applications of Hopf bifurcation. Cambridge University Press 1981 E. Hopf: Abzweigung einer periodischen Lösung von einer stationären Lösung eines ...

Author: Jan Awrejcewicz

Publisher: Springer Science & Business Media

ISBN: 9783642793295

Category: Science

Page: 272

View: 309

A collection of especially written articles describing the theory and application of nonlinear dynamics to a wide variety of problems encountered in physics and engineering. Each chapter is self-contained and includes an elementary introduction, an exposition of the state of the art, as well as details of recent theoretical, computational and experimental results. Included among the practical systems analysed are: hysteretic circuits, Josephson circuits, magnetic systems, railway dynamics, rotor dynamics and nonlinear dynamics of speech. This book provides important information and ideas for all mathematicians, physicists and engineers whose work in R & D or academia involves the practical consequences of chaotic dynamics.
Categories: Science