The Hopf Bifurcation and Its Applications / by J. E. Marsden, M. McCracken
(Applied Mathematical Sciences ; 19)
| データ種別 | 電子ブック |
|---|---|
| 出版者 | New York, NY : Springer New York |
| 出版年 | 1976 |
| 本文言語 | 英語 |
| 大きさ | 408 p : online resource |
書誌詳細を非表示
| 別書名 | 異なりアクセスタイトル:With contributions by numerous experts |
|---|---|
| 内容注記 | Section 1 Introduction to Stability and Bifurcation in Dynamical Systems and Fluid Dynamics Section 2 The Center Manifold Theorem Section 2A Some Spectral Theory Section 2B The Poincaré Map Section 3 The Hopf Bifurcation Theorem in R2 and in Rn Section 3A Other Bifurcation Theorems Section 3B More General Conditions for Stability Section 3C Hopf’s Bifurcation Theorem and the Center Theorem of Liapunov Section 4 Computation of the Stability Condition Section 4A How to use the Stability Formula; An Algorithm Section 4B Examples Section 4C Hopf Bifurcation and the Method of Averaging Section 5 A Translation of Hopf’s Original Paper Section 5A Editorial Comments Section 6 The Hopf Bifurcation Theorem Diffeomorphisms Section 6A The Canonical Form Section 7 Bifurcations with Symmetry Section 8 Bifurcation Theorems for Partial Differential Equations Section 8A Notes on Nonlinear Semigroups Section 9 Bifurcation in Fluid Dynamics and the Problem of Turbulence Section 9A On a Paper of G. Iooss Section 9B On a Paper of Kirchgässner and Kielhöffer Section 10 Bifurcation Phenomena in Population Models Section 11 A Mathematical Model of Two Cells Section 12 A Strange, Strange Attractor References |
| 一般注記 | 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 |
| 著者標目 | *Marsden, J. E. author McCracken, M. author SpringerLink (Online service) |
| 件 名 | LCSH:Mathematics LCSH:Matrix theory LCSH:Algebra FREE:Mathematics FREE:Linear and Multilinear Algebras, Matrix Theory |
| 分 類 | DC23:512.5 |
| 巻冊次 | ISBN:9781461263746 
|
| ISBN | 9781461263746 |
| URL | http://dx.doi.org/10.1007/978-1-4612-6374-6 |
目次／あらすじ