The Bianchi IX (mixmaster) cosmological model is not integrable

Latifi, A.; Musette, Micheline; Conte, Robert

doi:10.1016/0375-9601(94)00732-5

Cited by 65 publications

(67 citation statements)

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“…In short, Lyapunov exponents are not reliable indicators of chaos in general relativity. Using a different approach, it was shown that the mixmaster equations fail the Painlevé test [11]. This suggests that the mixmaster may be chaotic, but the Painlevé test is also inconclusive.…”

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confidence: 99%

The Mixmaster Universe is Chaotic

1997

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For the past decade there has been a considerable debate about the existence of chaos in the mixmaster cosmological model. The debate has been hampered by the coordinate, or observer, dependence of standard chaotic indicators such as Lyapunov exponents. Here we use coordinate-independent, fractal methods to show the mixmaster universe is indeed chaotic.[S0031-9007 (97)02322-3] PACS numbers: 98.80.Hw, 05.45. + bThe origin of the Universe and the fate of collapsing stars are two of the great mysteries in nature. In general relativity without exotic matter, the singularity theorems of Hawking and Penrose [1] argue that the gravitational collapse of very massive stars ends singular and that the Universe was born singular. These singular settings force gravity to face quantum mechanics. As well as exposing the fundamental laws of physics, the singular cores of black holes and the origin of the cosmos draw deep connections to the laws of thermodynamics and, as we will discuss here, to chaos.Earlier, Khalatnikov and Lifshitz [2] argued that singular solutions were the exception rather than the rule, putting them at odds with the singularity theorems. Their argument was that deformations in spacetime would be amplified during collapse and, consequently, would fight the formation of a singularity. This implied that the known symmetric singular solutions were atypical. The conflict was resolved when they realized the singularity in a collapsed star could be chaotic [3]. They conjectured that a generic singularity drives spacetime to churn and oscillate chaotically. Independently, Misner [4] suggested a chaotic approach to an early universe singularity. In his mixmaster universe, the different directions in three-space alternate in cycles of anisotropic collapse and expansion. A popular account of these developments may be found in Thorne's recent book [5].While the emergence of chaos helped our understanding of singularities in general relativity, building a resilient theory of relativistic chaos has become a task of its own. A debate has raged over whether or not the mixmaster universe is chaotic. Studies of the mixmaster dynamics using both approximate maps [6,7] and numerical integrations [8,9] have each yielded conflicting results as to the existence of positive Lyapunov exponents-a standard chaotic indicator. Finally, it was realized [9,10] that Lyapunov exponents are coordinate dependent and the conflicting results were a consequence of the different coordinate systems. In short, Lyapunov exponents are not reliable indicators of chaos in general relativity. Using a different approach, it was shown that the mixmaster equations fail the Painlevé test [11]. This suggests that the mixmaster may be chaotic, but the Painlevé test is also inconclusive.A detailed review of the mixmaster debate can be found in Ref. [12].In this Letter we show that the mixmaster universe is indeed chaotic by using coordinate independent, fractal methods. A fractal set of self-similar universes is uncovered by numerically solving Einstein's equ...

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confidence: 99%

The Mixmaster Universe is Chaotic

1997

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“…By a proper canonical reescaling of the variables in (11) and (19) it is easy to see that these constant energy surfaces are hyperspheres and that the constants of motion Q 1 , Q 2 and Q 3 = (E rot1 − E rot2 ) satisfy the algebra of the three dimensional rotation group under the Poisson bracket operation, namely,…”

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“…It is worth remarking that the linearization of this canonical transformation about the critical point E yields exactly the linear transformation (11), and that the variables (y, p y , z, p z ) correspond to the primed variables (K ′ , Y ′ , L ′ , Z ′ ) defined on the S 3 center-center manifold about a linear neighborhood of the critical point. The variable x is obviously the average scale factor of the model.…”

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“…the contributions to the Section Bianchi IX (Mixmaster) dynamics in [10], particularly [4]). Latifi et al [11] and Contopoulos et al [12] have shown the nonintegrability of the Mixmaster model in the Painlevé sense, although the question of the generic behaviour (chaotic or not) remained unsettled mainly due to the absence of an invariant (or topological) characterization of chaos in the model (standard chaotic indicators as Liapunov exponents being coordinate dependent and therefore questionable [13] [10]). More recently Cornish and Levin [14] proposed to quantify chaos in the Mixmaster universe by calculating the dimensions of fractal basin boundaries in initial conditions sets for the full dynamics, these boundaries being defined by the code associated with one of the three outcomes on which one of the three axes is collapsing most quickly, as established numerically.…”

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Homoclinic chaos in the dynamics of a general Bianchi type-IX model

et al. 2002

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The dynamics of a general Bianchi IX model with three scale factors is examined. The matter content of the model is assumed to be comoving dust plus a positive cosmological constant. The model presents a critical point of saddle-center-center type in the finite region of phase space. This critical point engenders in the phase space dynamics the topology of stable and unstable four dimensional tubes R × S 3 , where R is a saddle direction and S 3 is the manifold of unstable periodic orbits in the center-center sector. A general characteristic of the dynamical flow is an oscillatory mode about orbits of an invariant plane of the dynamics which contains the critical point and a Friedmann-Robertson-Walker (FRW) singularity. We show that a pair of tubes (one stable, one unstable) emerging from the neighborhood of the critical point towards the FRW singularity have homoclinic transversal crossings. The homoclinic intersection manifold has topology R × S 2 and is constituted of homoclinic orbits which are bi-asymptotic to the S 3 center-center manifold. This is an invariant signature of chaos in the model, and produces chaotic sets in phase space. The model also presents an asymptotic DeSitter attractor at infinity and initial conditions sets are shown to have fractal basin boundaries connected to the escape into the DeSitter configuration (escape into inflation), characterizing the critical point as a chaotic scatterer.The longtime debate on the chaotic dynamics of general Bianchi IX models started with the work of Belinskii, Khalatnikov and Lifshitz (BKL) on * Electronic address: henrique@fnal.gov † Electronic address: ozorio@cbpf.br ‡ Electronic address: ivano@cbpf.br § Electronic address: tonini@etfes.br the oscillatory behaviour of such models in their approach to the singularity [1]. They showed that the approach to the singularity(t → 0) of a general Bianchi IX cosmological solution is an oscillatory mode, consisting of an infinite sequence of periods (called Kasner eras) during which two of the scale functions oscillate and the third one decreases monotonically; on passing from one era to another the monotonic behaviour is transfered to another of the three scale functions. The length of

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“…The analytic integrability of Class A has been studied in previous works by several authors. [5][6][7][8][9][10][12][13][14][15][16][17] Here, our aim is to study the analytic integrability of all Bianchi models of class B in the variables ðx 1 ; x 2 ; x 3 ; x 4 ; x 5 ; x 6 Þ. In our study, we will use the following result, see Ref.…”

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On the absence of analytic integrability of the Bianchi Class B cosmological models

Ferragut

Llibre

Pantazi

2013

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We follow Bogoyavlensky's approach to deal with Bianchi class B cosmological models. We characterize the analytic integrability of such systems. V C 2013 American Institute of Physics.[http://dx.doi.org/10.1063/1.4790828]Bianchi models are cosmological models that describe space-times which are foliated by homogeneous hypersurfaces of constant time and are divided into two classes, Class A and Class B. There are many studies about the integrability of Class A. Here, we study the integrability of Class B. For the homogeneous cosmological models of Class B, Einstein's system of differential equations reduces to a dynamical system of dimension seven according to Bogoyavlensky's approach. We show that in order to study the integrability of such systems, it is sufficient to deal with homogeneous polynomial differential systems of dimension six. Concretely, Bianchi V is the simplest model and can be written as a homogeneous polynomial differential system of degree 2. Bianchi IV is dealt as a homogeneous polynomial differential system of degree 3 and the rest of the models, Bianchis III, VI, and VII are of degree 5. Due to the fact that all Bianchi class B models have been reduced to homogeneous polynomial differential systems, the study of their analytic integrability reduces to analyze their homogeneous polynomial first integrals. We show that Bianchi model V admits polynomial first integral, and we prove that the corresponding homogeneous polynomial differential systems that represent models Bianchi IV, III, VI, and VII do not admit polynomial first integrals. The fact that these Bianchi models are not completely integrable with analytic first integrals facilitates that they can have chaotic behavior.

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The Bianchi IX (mixmaster) cosmological model is not integrable

Cited by 65 publications

References 34 publications

The Mixmaster Universe is Chaotic

The Mixmaster Universe is Chaotic

Homoclinic chaos in the dynamics of a general Bianchi type-IX model

On the absence of analytic integrability of the Bianchi Class B cosmological models

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