Bifurcation without parameters

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.

Pt. I Preliminaries -- 1.Introduction -- 1.1.Classical Bifurcation Versus Bifurcation Without Parameters -- 1.2.Manifolds of Equilibria -- 1.2.1.Conserved Quantities -- 1.2.2.Equivariances -- 1.2.3.Reversibilities -- 1.2.4.Singular Perturbations -- 1.2.5.Perturbing the Manifold -- 1.2.6.Cosymmetries -- 1.3.Classification of Bifurcation Types -- 1.3.1.Codimension One -- 1.3.2.Codimension Two -- 1.4.Further Cases -- 2.Methods and Concepts -- 2.1.Center Manifolds -- 2.2.Normal Forms -- 2.3.Normal Forms with Manifolds of Equilibria -- 2.4.Genericity -- 2.5.Unfoldings and Codimension -- 2.6.Rescaling and Blow Up -- 3.Cosymmetries -- pt. II Codimension One -- 4.Transcritical Bifurcation -- 4.1.The Generic Case -- 4.2.Additional Reflection Symmetry -- 5.Poincare-Andronov-Hopf Bifurcation -- 6.Application: Decoupling in Networks -- 7.Application: Oscillatory Profiles in Systems of Hyperbolic Balance Laws -- pt. III Codimension Two -- 8.Degenerate Transcritical Bifurcation -- 8.1.Families of Lines of Equilibria: Singular Drift -- 8.2.Families of Lines of Equilibria: Fold -- 8.3.Planes of Equilibria -- 9.Degenerate Poincare-Andronov-Hopf Bifurcation -- 9.1.Families of Lines of Equilibria: S…