Liebscher S. Bifurcation without Parameters

N.-Y.: Springer, 2015. — 158 p.
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 and Concepts.
Cosymmetries.Transcritical Bifurcation.
Poincaré-Andronov-Hopf Bifurcation.
Application: Decoupling in Networks.
Application: Oscillatory Profiles in Systems of Hyperbolic Balance Laws.Degenerate Transcritical Bifurcation.
Degenerate Poincaré-Andronov-Hopf Bifurcation.
Bogdanov-Takens Bifurcation.
Zero-Hopf Bifurcation.
Double-Hopf Bifurcation.
Application: Cosmological Models of Bianchi Type, the Tumbling Universe.
Application: Fluid Flow in a Planar Channel, Spatial Dynamics with Reversible Bogdanov-Takens Bifurcation.Codimension-One Manifolds of Equilibria.
Summary and Outlook.