Bifurcation and fine structure phenomena in critical collapse of a self-gravitating σ-field

Aichelburg, P. C.; Bizoń, Piotr; Tabor, Zbisław

doi:10.1088/0264-9381/23/16/s01

Cited by 6 publications

(6 citation statements)

References 16 publications

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“…One of them is the existence of Hopf bifurcations where a self-similar solution (a stable fixed point) is transformed into a discrete self-similar solution (limit cycle) as a certain parameter varies (see [49]). Other kinds of bifurcations, for example of the Shilnikov type, are found as well [50]. Now we demonstrate that chaotic behaviour is also possible.…”

Section: Strange Attractors and Exotic Behavioursupporting

confidence: 67%

A catalogue of singularities

Eggers,

Fontelos

2007

Preprint

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This paper is an attempt to classify finite-time singularities of PDEs. Most of the problems considered describe free-surface flows, which are easily observed experimentally. We consider problems where the singularity occurs at a point, and where typical scales of the solution shrink to zero as the singularity is approached. Upon a similarity transformation, exact self-similar behaviour is mapped to the fixed point of a infinite dimensional dynamical system representing the original dynamics. We show that the dynamics close to the fixed point is a useful way classifying the structure of the singularity. Specifically, we consider various types of stable and unstable fixed points, centre-manifold dynamics, limit cycles, and chaotic dynamics.

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Section: Strange Attractors and Exotic Behavioursupporting

confidence: 67%

A catalogue of singularities

Eggers,

Fontelos

2007

Preprint

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“…High-precision numerics in [3] further supports this picture: For η > η c a codimension-1 CSS solution coexists in phase space with a codimension-1 DSS attractor such that the (1-dimensional) unstable manifold of the DSS solution lies on the stable manifold of the CSS solution. For η close to η c the two solutions are close and the orbits around the DSS solution become slower because they spend more time in the neighbourhood of the CSS attractor.…”

Section: More Spherical Symmetrymentioning

confidence: 91%

Critical Phenomena in Gravitational Collapse

Gundlach

Martín-García

2007

Living Rev. Relativ.

285

266

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As first discovered by Choptuik, the black hole threshold in the space of initial data for general relativity shows both surprising structure and surprising simplicity. Universality, power-law scaling of the black hole mass, and scale echoing have given rise to the term “critical phenomena”. They are explained by the existence of exact solutions which are attractors within the black hole threshold, that is, attractors of codimension one in phase space, and which are typically self-similar. Critical phenomena give a natural route from smooth initial data to arbitrarily large curvatures visible from infinity, and are therefore likely to be relevant for cosmic censorship, quantum gravity, astrophysics, and our general understanding of the dynamics of general relativity.

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“…One of them is the existence of Hopf bifurcations where a self-similar solution (a stable fixed point) is transformed into a discrete self-similar solution (limit cycle) as a certain parameter varies (see [143]). Other kinds of bifurcations, for example of the Shilnikov type, are found as well [144]. Before coming to simple explicit examples, we mention that possible complex dynamics in τ has long been suggested for simplified versions of the inviscid Euler equations [145,146,141].…”

Section: Strange Attractors and Exotic Behaviourmentioning

confidence: 95%

The role of self-similarity in singularities of partial differential equations

2008

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We survey rigorous, formal, and numerical results on the formation of point-like singularities (or blow-up) for a wide range of evolution equations. We use a similarity transformation of the original equation with respect to the blow-up point, such that self-similar behaviour is mapped to the fixed point of a dynamical system. We point out that analysing the dynamics close to the fixed point is a useful way of characterising the singularity, in that the dynamics frequently reduces to very few dimensions. As far as we are aware, examples from the literature either correspond to stable fixed points, low-dimensional centre-manifold dynamics, limit cycles, or travelling waves. For each "class" of singularity, we give detailed examples.

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Bifurcation and fine structure phenomena in critical collapse of a self-gravitating σ-field

Cited by 6 publications

References 16 publications

A catalogue of singularities

A catalogue of singularities

Critical Phenomena in Gravitational Collapse

The role of self-similarity in singularities of partial differential equations

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