Targeted at mathematicians having at least a basic familiarity with classical bifurcation theory, this monograph provides a systematic classification and analysis of bifurcations without parameters in dynamical systems. Although the methods and concepts are briefly introduced, a prior knowledge of center-manifold reductions and normal-form calculations will help the reader to appreciate the presentation. Bifurcations without parameters occur along manifolds of equilibria, at points where normal hyperbolicity of the manifold is violated. The general theory, illustrated by many applications, aims at a geometric understanding of the local dynamics near the bifurcation points.
Author(s): Stefan Liebscher (auth.)
Series: Lecture Notes in Mathematics 2117
Edition: 1
Publisher: Springer International Publishing
Year: 2015
Language: English
Pages: 142
Tags: Ordinary Differential Equations; Partial Differential Equations; Dynamical Systems and Ergodic Theory
Front Matter....Pages i-xii
Front Matter....Pages 1-1
Introduction....Pages 3-12
Methods and Concepts....Pages 13-19
Cosymmetries....Pages 21-23
Front Matter....Pages 25-25
Transcritical Bifurcation....Pages 27-34
Poincaré-Andronov-Hopf Bifurcation....Pages 35-41
Application: Decoupling in Networks....Pages 43-47
Application: Oscillatory Profiles in Systems of Hyperbolic Balance Laws....Pages 49-54
Front Matter....Pages 55-55
Degenerate Transcritical Bifurcation....Pages 57-65
Degenerate Poincaré-Andronov-Hopf Bifurcation....Pages 67-79
Bogdanov-Takens Bifurcation....Pages 81-102
Zero-Hopf Bifurcation....Pages 103-108
Double-Hopf Bifurcation....Pages 109-113
Application: Cosmological Models of Bianchi Type, the Tumbling Universe....Pages 115-118
Application: Fluid Flow in a Planar Channel, Spatial Dynamics with Reversible Bogdanov-Takens Bifurcation....Pages 119-128
Front Matter....Pages 129-129
Codimension-One Manifolds of Equilibria....Pages 131-133
Summary and Outlook....Pages 135-137
Back Matter....Pages 139-144
