Titlebook: Bifurcation without Parameters; Stefan Liebscher Book 2015 The Editor(s) (if applicable) and The Author(s), under exclusive license to Spr

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OGLE

Intuitive

諂媚于人

Degenerate Transcritical Bifurcation. At isolated points, one of the non-degeneracy conditions?(4.8,?4.9) may fail and codimension-two singularities appear. We shall discuss these degeneracies, first in a one-parameter-family of lines of equilibria and then along a two-dimensional equilibrium surface.

etiquette

Zero-Hopf Bifurcationsverse to a plane of equilibria. It turns out that instead we can study a one-parameter family of lines in a system depending on one parameter. Indeed, the rescaled normal form?(11.6) is the same in both cases.

包租車船

Double-Hopf Bifurcationof equilibria. As we shall see, the drift direction at the Hopf lines play an important role. In the case of a parameter-dependent fixed line of equilibria, drifts at both Hopf-lines can be opposite and spiraling orbits appear, see Sect. 12.1. In the generic case with a plane of equilibria without p

indifferent

Application: Cosmological Models of Bianchi Type, the Tumbling Universe—to the matter content. Bianchi models, in particular, are homogeneous but anisotropic solutions. They are given by a five-dimensional ODE system in expansion-reduced variables and feature the Kasner circle . of equilibria and caps filled with heteroclinic orbits connecting equilibria on .. On the K

飲料

Application: Fluid Flow in a Planar Channel, Spatial Dynamics with Reversible Bogdanov-Takens Bifurcreduction has been used to find time-independent bounded solutions at the onset of instability of the system when the Reynolds number increases. We regard bounded solutions as evolutions in the unbounded direction of a cross-sectional profile, and find a six-dimensional center manifold. Three conser

AMEND

Codimension-One Manifolds of Equilibriation also in Chaps. 5, 9, we removed the manifold of equilibria by multiplying with a singular factor 1∕. or 1∕.. This idea required that there is only one transverse direction to the manifold of equilibria. For such manifolds of codimension one, in phase space, we can generalize the idea.

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0075-8434 heory, illustrated by many applications, aims at a geometric understanding of the local dynamics near the bifurcation points.978-3-319-10776-9978-3-319-10777-6Series ISSN 0075-8434 Series E-ISSN 1617-9692