Homoclinic chaos and energy condition violation

Heinzle, Jakob; Röhr, Niklas; Uggla, Claes

doi:10.1103/physrevd.74.061502

“…The latter has naturally been examined in cosmological solutions, mainly in Bianchi IX (Mixmaster) models. While the secular debate about chaoticity and the very nature of these universes continues (Coley 2002; Fay & Lehner 2004; Benini & Montani 2004; Soares & Stuchi 2005; Heinzle, Röhr & Uggla 2006; Buzzi, Llibre & da Silva 2007; Andriopoulos & Leach 2008; Heinzle & Uggla 2009), one of the later attractors is to assess the effect of the cosmological constant and/versus that of a scalar field within the Friedmann–Lematre–Robertson–Walker model dynamics (Jorás & Stuchi 2003; Faraoni, Jensen & Theuerkauf 2006; Hrycyna & Szydłowski 2006; Lukes‐Gerakopoulos, Basilakos & Contopoulos 2008; Maciejewski et al 2008). [We only give several more recent references here.…”

Section: Introductionmentioning

confidence: 99%

Free motion around black holes with discs or rings: between integrability and chaos - I

Semerák¹,

Suková²

2010

Monthly Notices of the Royal Astronomical Society

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Geodesic dynamics is regular in the fields of isolated stationary black holes. However, due to the presence of unstable periodic orbits, it easily becomes chaotic under various perturbations. Here, we examine what amount of stochasticity is induced in Schwarzschild space–time by the presence of an additional source. Following astrophysical motivation, we specifically consider thin rings or discs lying symmetrically around the hole, and describe the total field in terms of exact static and axially symmetric solutions of Einstein's equations. The growth of chaos in time‐like geodesic motion is illustrated on Poincaré sections, on time series of position or velocity and their Fourier spectra, and on time evolution of the orbital ‘latitudinal action’. The results are discussed in terms of dependence on the mass and position of the ring/disc and on geodesic parameters (energy and angular momentum). In the Introduction, we also add an overview of the literature.

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“…Moreover, theorems about the global asymptotic behavior of solutions was established. Results in [113,114] disprove claims of non-predictability and chaos for models close to Einstein's model given in [115][116][117][118].…”

Section: Closed Flrw: Regime >mentioning

confidence: 51%

“…Thus, a set of models in a neighborhood of Einstein's model were claimed to be unpredictable and characterized by "homoclinic chaos" [115][116][117][118]. However, the asserted "homoclinic phenomena", if they occur at all, must be confined to narrow regions of the phase space [113] (see also [114]). Now, we summarize the results of the "Averaging generalized scalar-field cosmologies" research program.…”

Section: Discussionmentioning

confidence: 99%

Averaging generalized scalar-field cosmologies III: Kantowski–Sachs and closed Friedmann–Lemaître–Robertson–Walker models

León

¹

,

González

²

,

Lepe

³

et al. 2021

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Scalar-field cosmologies with a generalized harmonic potential and matter with energy density $$\rho _m$$ ρ m , pressure $$p_m$$ p m , and barotropic equation of state (EoS) $$p_m=(\gamma -1)\rho _m, \; \gamma \in [0,2]$$ p m = ( γ - 1 ) ρ m , γ ∈ [ 0 , 2 ] in Kantowski–Sachs (KS) and closed Friedmann–Lemaître–Robertson–Walker (FLRW) metrics are investigated. We use methods from non-linear dynamical systems theory and averaging theory considering a time-dependent perturbation function D. We define a regular dynamical system over a compact phase space, obtaining global results. That is, for KS metric the global late-time attractors of full and time-averaged systems are two anisotropic contracting solutions, which are non-flat locally rotationally symmetric (LRS) Kasner and Taub (flat LRS Kasner) for $$0\le \gamma \le 2$$ 0 ≤ γ ≤ 2 , and flat FLRW matter-dominated universe if $$0\le \gamma \le \frac{2}{3}$$ 0 ≤ γ ≤ 2 3 . For closed FLRW metric late-time attractors of full and averaged systems are a flat matter-dominated FLRW universe for $$0\le \gamma \le \frac{2}{3}$$ 0 ≤ γ ≤ 2 3 as in KS and Einstein–de Sitter solution for $$0\le \gamma <1$$ 0 ≤ γ < 1 . Therefore, a time-averaged system determines future asymptotics of the full system. Also, oscillations entering the system through Klein–Gordon (KG) equation can be controlled and smoothed out when D goes monotonically to zero, and incidentally for the whole D-range for KS and closed FLRW (if $$0\le \gamma < 1$$ 0 ≤ γ < 1 ) too. However, for $$\gamma \ge 1$$ γ ≥ 1 closed FLRW solutions of the full system depart from the solutions of the averaged system as D is large. Our results are supported by numerical simulations.

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“…It would be of interest to perform numerical computations for these more general f (t, h k ). Second, our system was simplified by assuming a diagonal metric (41), and it could be generalized to include off-diagonal elements and possibly even rotating models. Third, our examples focused on massless cases for simplicity, and the massive case should be investigated.…”

Section: Discussionmentioning

confidence: 99%

Hamiltonian dynamics of spatially-homogeneous Vlasov-Einstein systems

Okabe

¹

,

Morrison

²

,

Friedrichsen

³

et al. 2011

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We introduce a new matter action principle, with a wide range of applicability, for the Vlasov equation in terms of a conjugate pair of functions. Here we apply this action principle to the study of matter in Bianchi cosmological models in general relativity. The Bianchi models are spatially-homogeneous solutions to the Einstein field equations, classified by the three-dimensional Lie algebra that describes the symmetry group of the model. The Einstein equations for these models reduce to a set of coupled ordinary differential equations. The class A Bianchi models admit a Hamiltonian formulation in which the components of the metric tensor and their time derivatives yield the canonical coordinates. The evolution of anisotropy in the vacuum Bianchi models is determined by a potential due to the curvature of the model, according to its symmetry.For illustrative purposes, we examine the evolution of anisotropy in models with Vlasov matter.The Vlasov content is further simplified by the assumption of cold, counter-streaming matter, a kind of matter that is far from thermal equilibrium and is not describable by an ordinary fluid model nor other more simplistic matter models. Qualitative differences and similarities are found in the dynamics of certain vacuum class A Bianchi models and Bianchi Type I models with cold, counter-streaming Vlasov matter potentials analogous to the curvature potentials of corresponding vacuum models.

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Homoclinic chaos and energy condition violation

Cited by 4 publications

References 15 publications

Free motion around black holes with discs or rings: between integrability and chaos - I

Free motion around black holes with discs or rings: between integrability and chaos - I

Averaging generalized scalar-field cosmologies III: Kantowski–Sachs and closed Friedmann–Lemaître–Robertson–Walker models

Hamiltonian dynamics of spatially-homogeneous Vlasov-Einstein systems

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