Invariant Lyapunov exponents and chaos in cosmology

Abstract

The precise dynamical nature of the Bianchi type-IX cosmological model has been a subject of debate for a number of years. The lack of congruence between various results, which mainly employ numerical calculations of the spectrum of Lyapunov exponents, has motivated a number of attempts at a gaugeinvariant characterization of chaos. Here we critically examine one of these attempts and argue that none of the attempted characterizations so far put forward are completely satisfactory. We also briefly discuss natural gauges in cosmology and distinguish between qualitative and quantitative measures of chaos.PACS number(s1: 98.80.H~

Chaos and Fractals Around Black Holes

“…Other conditions, such as a compact phase space and mixing of trajectories are required to prove that a system exhibits chaos [19]. This method has the advantage that it is based on curvature scalars which do not depend on a choice of coordinate system, however its applicability to the Mixmaster system has been questioned [20]. There has also been a study of the homoclinic loops of a Mixmaster Universe containing matter, albeit with an unrealistic equation of state [21].…”

Chaos in pseudo-Newtonian black holes with halos

“…We always found LCN < 5 10 −5 . The Lyapunov-like coefficients used in general relativistic systems may have different forms, such as the one suggested by Burd & Tavakol (1993) in the study of Bianchi IX systems. However, we have studied a simple system with no singularities other than the black hole, where we have a well-defined evolution parameter.…”

Covariance of the mixmaster chaoticity

“…It is worth noting that the success of our analysis, in showing the time gauge independence of the Mixmaster chaos, relies on the use of a standard ADM reduction of the variational principle (which reduces the system by one degree of freedom) and overall because, adopting Misner-Chitré-like variables, the asymptotic potential walls are fixed in time. The difference between our approach and the one presented in [23] (see also for a critical analysis [24]) consists effectively in these features, though in those works is even faced the problem of the Mixmaster chaos covariance with respect to the choice of generic configuration variables.…”