3,020 research outputs found
Statistical properties of Lorenz like flows, recent developments and perspectives
We comment on mathematical results about the statistical behavior of Lorenz
equations an its attractor, and more generally to the class of singular
hyperbolic systems. The mathematical theory of such kind of systems turned out
to be surprisingly difficult. It is remarkable that a rigorous proof of the
existence of the Lorenz attractor was presented only around the year 2000 with
a computer assisted proof together with an extension of the hyperbolic theory
developed to encompass attractors robustly containing equilibria. We present
some of the main results on the statisitcal behavior of such systems. We show
that for attractors of three-dimensional flows, robust chaotic behavior is
equivalent to the existence of certain hyperbolic structures, known as
singular-hyperbolicity. These structures, in turn, are associated to the
existence of physical measures: \emph{in low dimensions, robust chaotic
behavior for flows ensures the existence of a physical measure}. We then give
more details on recent results on the dynamics of singular-hyperbolic
(Lorenz-like) attractors.Comment: 40 pages; 10 figures; Keywords: sensitive dependence on initial
conditions, physical measure, singular-hyperbolicity, expansiveness, robust
attractor, robust chaotic flow, positive Lyapunov exponent, large deviations,
hitting and recurrence times. Minor typos corrected and precise
acknowledgments of financial support added. To appear in Int J of Bif and
Chaos in App Sciences and Engineerin
Tilted two-fluid Bianchi type I models
In this paper we investigate expanding Bianchi type I models with two tilted
fluids with the same linear equation of state, characterized by the equation of
state parameter w. Individually the fluids have non-zero energy fluxes w.r.t.
the symmetry surfaces, but these cancel each other because of the Codazzi
constraint. We prove that when w=0 the model isotropizes to the future. Using
numerical simulations and a linear analysis we also find the asymptotic states
of models with w>0. We find that future isotropization occurs if and only if . The results are compared to similar models investigated previously
where the two fluids have different equation of state parameters.Comment: 14 pages, 3 figure
Direct transition to high-dimensional chaos through a global bifurcation
In the present work we report on a genuine route by which a high-dimensional
(with d>4) chaotic attractor is created directly, i.e., without a
low-dimensional chaotic attractor as an intermediate step. The high-dimensional
chaotic set is created in a heteroclinic global bifurcation that yields an
infinite number of unstable tori.The mechanism is illustrated using a system
constructed by coupling three Lorenz oscillators. So, the route presented here
can be considered a prototype for high-dimensional chaotic behavior just as the
Lorenz model is for low-dimensional chaos.Comment: 7 page
Hyperinflation generalised: from its attractor mechanism to its tension with the `swampland conjectures'
In negatively curved field spaces, inflation can be realised even in steep
potentials. Hyperinflation invokes the `centrifugal force' of a field orbiting
the hyperbolic plane to sustain inflation. We generalise hyperinflation by
showing that it can be realised in models with any number of fields
(), and in broad classes of potentials that, in particular, don't
need to be rotationally symmetric. For example, hyperinflation can follow a
period of radial slow-roll inflation that undergoes geometric destabilisation,
yet this inflationary phase is not identical to the recently proposed scenario
of `side-tracked inflation'. We furthermore provide a detailed proof of the
attractor mechanism of (the original and generalised) hyperinflation, and
provide a novel set of characteristic, explicit models. We close by discussing
the compatibility of hyperinflation with observations and the recently much
discussed `swampland conjectures'. Observationally viable models can be
realised that satisfy either the `de Sitter conjecture' () or
the `distance conjecture' (), but satisfying both
simultaneously brings hyperinflation in some tension with successful reheating
after inflation. However, hyperinflation can get much closer to satisfying all
of these criteria than standard slow-roll inflation. Furthermore, while the
original model is in stark tension with the weak gravity conjecture,
generalisations can circumvent this issue.Comment: 26 pages, 3 figure
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