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

谄媚于人

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.

包租车船

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

饮料

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.

内向者

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