Bifurcation without Parameters
Differential Equations Differential equations, partial Differentiable dynamical systems Ordinary Differential Equations Partial Differential Equations Dynamical Systems and Ergodic Theory
Imprint: Springer
2015
1st ed. 2015.
EISBN 3319107771
Introduction.
Methods & Concepts.
Cosymmetries.
Codimension One.
Transcritical Bifurcation.
Poincar´e-Andronov-Hopf Bifurcation.
Application: Decoupling in Networks.
Application: Oscillatory Profiles.
Codimension Two.
egenerate Transcritical Bifurcation.
egenerate Andronov-Hopf Bifurcation.
Bogdanov-Takens Bifurcation.
Zero-Hopf Bifurcation.
Double-Hopf Bifurcation.
Application: Cosmological Models.
Application: Planar Fluid Flow.
Beyond Codimension Two.
Codimension-One Manifolds of Equilibria.
Summary & Outlook.
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.
Methods & Concepts.
Cosymmetries.
Codimension One.
Transcritical Bifurcation.
Poincar´e-Andronov-Hopf Bifurcation.
Application: Decoupling in Networks.
Application: Oscillatory Profiles.
Codimension Two.
egenerate Transcritical Bifurcation.
egenerate Andronov-Hopf Bifurcation.
Bogdanov-Takens Bifurcation.
Zero-Hopf Bifurcation.
Double-Hopf Bifurcation.
Application: Cosmological Models.
Application: Planar Fluid Flow.
Beyond Codimension Two.
Codimension-One Manifolds of Equilibria.
Summary & Outlook.
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.
