Marek Szydlowski

Structurally stable approximations to Friedmann—Lemaître world models

Friedmann—Lemaître cosmology is briefly reviewed in terms of dynamical systems. It is demonstrated that in certain cases bulk viscosity dissipation structurally stabilizes Friedmann—Lemaître solutions. It turns out that, for A=0, there are structurally stable solutions if ξ~ε1/2, where ξ is the bulk viscosity coefficient. For A≠0 structurally stable solutions are essentially those with ξ=const. The role of structural stability in physics and cosmology is shortly discussed.

Structural stability properties of friedman cosmology

A dynamical system with Robertson-Walker symmetries and the equation of the statep=γ∈, 0≤γ≤1, considered both as a conservative and nonconservative system, is studied with respect to its structural stability properties. Different cases are shown and analyzed on the phase space (x=RD, y=x).

The Eisenhart Geometry as an Alternative Description of Dynamics in Terms of Geodesics

We show the advantages of representing the dynamics of simple mechanical systems, described by a natural Lagrangian, in terms of geodesics of a Riemannian (or pseudo-Riemannian) space with an additional dimension. We demonstrate how trajectories of simple mechanical systems can be put into one-to-one correspondence with the geodesics of a suitable manifold. Two different ways in which geometry of the configuration space can be obtained from a higher dimensional model are presented and compared: First, by a straightforward projection, and second, as a space geometry of a quotient space obtained by the action of the timelike Killing vector generating a stationary symmetry of a background space geometry with an additional dimension. The second model is more informative and coincides with the so-called optical model of the line of sight geometry. On the base of this model we study the behaviour of nearby geodesics to detect their sensitive dependence on initial conditions—the key ingredient of deterministic chaos. The advantage of such a formulation is its invariant character.

Tolman's cosmological models

Oscillating Tolman universes have been identified as solutions of the Friedmann equation with the bulk viscosity dissipation. Their dynamics and some thermodynamical properties are briefly discussed. Problems such as the Eternal Return philosophy and its modern incarnation into the doctrine of the Universe as a global oscillator, Tiplers no-return theorem, etc., are touched upon.

Dynamical dimensional reduction

We propose to call a dynamical dimensional reduction effective if the corresponding dynamical system possesses a single attracting critical point representing expanding physical space-time and static internal space. We show that theBV × TD multidimensional cosmological model with a hydrodynamic energy-momentum tensor provides an example of effective dimensional reduction. We also study the dynamics of the multidimensional cosmological model of typeBI × TD with an energy-momentum tensor representing low temperature quantum effects, monopole contribution and the cosmological constant. It turns out that anisotropy and the cosmological constant are crucial for the process of dimensional reduction to be effective. We argue that this is the general property of homogeneous multidimensional cosmological models.

Generic and nongeneric world models

Catastrophe theory methods are employed to obtain a new classification of those world models which can be presented in the form of gradient dynamical systems. Generic sets and structural stability of models in the potential space are strictly defined. It is shown that if a cosmological model is required to be Friedman and generic, it must be flat.

Chaos Hidden Behind Time Parametrization in the Mixmaster Cosmology

We define the Maupertuis clock which counts Kasner epochs in the Mixmaster cosmology. The characteristic time scale as measured by this clock is just the length of a given Kasner epoch. We show that in every transition from one Kasner epoch to another one unit of information is lost. Relationships of the Maupertuis time with other time parametrizations used in general relativity (cosmological time, Misner time, Chitre-Misner time, curvature time and superspace time) are investigated. In the logarithmic (mechanical) time nearby trajectories diverge linearly, and the system behaves as if it were integrable and the chaos is “hidden” behind this parametrization. The physical meaning of the Maupertuis time, as a chaos indicator, is discussed. We also investigate the dependence of the Lyapunov exponents on time reparametrizations.

Bianchi A Cosmological Models as the Simplest Dynamical Systems in R4

The behaviour of the vacuum non-tilted Bianchimodels of class A is studied in terms of dynamicalsystems theory. We introduce phase variables in whichthe Hamiltonian constraint is solved algebraically. It is shown that in these variables BianchiVIII and Bianchi IX models assume the form of afour-dimensional autonomous system with a polynomialvector field defined on the phase space, whereas Bianchi I and Bianchi II world models can be presentedas a one- and two-dimensional system, respectively. TheBianchi VI0 and Bianchi VII0 worldmodels are represented as a three-dimensional dynamicalsystem.

The local instability of mixmaster dynamical systems

In the present work the connection between chaotic behaviour and dimensions of space-time for Mixmaster models is discussed.

Are Einstein's equations of homogeneous multidimensional cosmology generic?

It is shown that Einsteins equations for multidimensional homogeneous block diagonal cosmology are generic only if the dimension of space-time is 4.