Giovanni Imponente

Classical and Quantum Features of the Mixmaster Singularity

This review paper is devoted to analyzing the main properties of the cosmological singularity associated with the homogeneous and inhomogeneous Mixmaster model. After the introduction of the main tools required to treat the cosmological issue, we review in detail the main results obtained over the last forty years on the Mixmaster topic. We first assess the classical picture of the homogeneous chaotic cosmologies and, after a presentation of the canonical method for the quantization, we develop the quantum Mixmaster behavior. Finally, we extend both the classical and the quantum features to the fully inhomogeneous case. Our survey analyzes the fundamental framework of the Mixmaster picture and completes it by accounting for recent and peculiar outstanding results.

Complex dynamics of the biological rhythms: gallbladder and heart cases

A theoretical analysis of the mechanisms underlying the dynamics of gallbladder and heart pulsation could clarify the question regarding the classification as chaotic of the associated behaviour, eventually related to a normal and healthy beat; this analysis is particularly relevant in view of the control of dynamic bifurcations arising in situations of disease. In this work a summary of the DFA method applied to gallbladder volume data for a modest number of healthy and ill patients is presented: the presence of signal correlation is found in both cases, but the fit shapes differ from some critical values.

COVARIANT LYAPUNOV EXPONENTS FOR THE MIXMASTER

The dynamics of the Mixmaster Universe is analized in a covariant picture via Misner--Chitre-like variables for an ADM Hamiltonian approach. The system outcomes as isomorphic to a billiard on the Lobachevsky plane and Lyapunov exponents are calculated explicitly.

Bianchi IX chaoticity: BKL map and continuous flow

We analyze the Bianchi IX dynamics (Mixmaster) in view of its stochastic properties; in the present paper we address either the original approach due to Belinski, Khalatnikov and Lifshitz (BKL) as well as a Hamiltonian one relying on the Arnowitt–Deser–Misner (ADM) reduction.

INHOMOGENEOUS de SITTER SOLUTION WITH SCALAR FIELD AND PERTURBATIONS SPECTRUM

We provide an inhomogeneous solution concerning the dynamics of a real self-interacting scalar field minimally coupled to gravity in a region of the configuration space where it performs a slow rolling on a plateau of its potential. During the inhomogeneous de Sitter phase the scalar field dominant term is a function of the spatial coordinates only. This solution is near to the FLRW model allows a classical origin for the inhomogeneous perturbations spectrum.

MIXMASTER CHAOTICITY AS SEMICLASSICAL LIMIT OF THE CANONICAL QUANTUM DYNAMICS

Within a cosmological framework, we provide a Hamiltonian analysis of the Mixmaster Universe dynamics on the base of a standard Arnowitt–Deser–Misner approach, showing how the chaotic behavior characterizing the evolution of the system near the cosmological singularity can be obtained as the semiclassical limit of the canonical quantization of the model in the same dynamical representation. The relation between this intrinsic chaotic behavior and the indeterministic quantum dynamics is inferred through the coincidence between the microcanonical probability distribution and the semiclassical quantum one.

On the quantum origin of the Mixmaster chaos covariance

Abstract Our analysis shows how the covariant chaotic behaviour characterizing the evolution of the Mixmaster cosmology near the initial singularity can be taken as the semiclassical limit in the canonical quantization performed by the corresponding Hamiltonian representation.

MIXMASTER CHAOS VIA THE INVARIANT MEASURE

The chaoticity of the Mixmaster is discussed in the framework of Statistical Mechanics by using Misner--Chitre-like variables and an ADM reduction of its dynamics. We show that such a system is well described by a microcanonical ensemble whose invariant measure is induced by the corresponding Liouville one and is uniform. The covariance with respect to the choice of the temporal gauge of the obtained invariant measure is outlined.