25. M. Demianski, Z. A. Golda, M. Heller and M. Szydłowski
Kantowski-Sachs multidimensional cosmological models and dynamical dimensional reduction
Class. Quantum Grav., vol. 5, pp. 733-742 (1988).
[abstract] [journal]

Abstract:
Einstein's field equations are solved for a multidimensional spacetime $(KS)\times T^m$, where ($KS$) is a four-dimensional Kantowski-Sachs' spacetime (1966) and $T^m$ is an $m$-dimensional torus. Among all possible vacuum solutions there is a large class of spacetimes in which the macroscopic space expands and the microscopic space contracts to a finite volume. The authors also consider a non-vacuum case and they explicitly solve the field equations for the matter satisfying the Zel'dovich equations of state (1987). In non-vacuum models, with matter satisfying an equation of state $\rho = \gamma\rho$. $0\leq \gamma\leq 1$, at a sufficiently late stage of evolution the microspace always expands and the dynamical dimensional reduction does not occur. Models $(KS)\times B(IX)\times S^1\times S^1\times S^1\times S^1$ and $(KS)\times B(IX)\times B(IX)\times S^1$, where $B(IX)$ is the Bianchi type-$IX$ space, are also briefly discussed. It is shown that there is no chaotic behaviour in these cases. The same conclusion is also valid when one-loop high-temperature quantum corrections generated by a massless scalar field are taken into account.

26. Zdzislaw A. Golda, Marek Szydlowski, Michal Heller
Generic and nongeneric world models
Gen. Rel. Grav., vol. 19, pp. 707-718 (1987).
[abstract] [journal]

Abstract:
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.

27. M. Demianski, Z. A. Golda, M. Heller and M. Szydlowski
The dimensional reduction in a multi-dimensional cosmology
Class. Quantum Grav. , vol. 3, pp. 1199-1205 (1986).
[abstract] [journal]

Abstract:
Einstein's field equation are solved for the case of the eleven-dimensional vacuum spacetime which is the product $R\times\mbox{Bianchi}(V)\times T^7$, where $T^7$ is a seven-dimensional torus. Among all possible solutions the authors identify those in which the macroscopic space expands and the microscopic space contracts to a finite size. The solutions with this property are `typical' within the considered class. They implement the idea of a purely dynamical dimensional reduction.

28. M. Szydłowski, M. Heller, Z. Golda
Stochastic Time Scale for the Universe
Acta Phys. Pol. , vol. B17, pp. 19-24 (1986).
[abstract]

Abstract:
An intrinsic time scale is naturally defined within stochastic gradient dynamical systems. It should be interpreted as a ``relaxation time'' to a local potential minimum after the system bas been randomly perturbed. It is shown that for a flat Friedman-like cosmological model this time scale is of order of the age of the Universe.

29. M. Szydłowski, M. Heller, Z. Golda
Stochastic Properties of the Friedman Dynamical System
Acta Phys. Pol. , vol. B16, pp. 791-798 (1985).
[abstract]

Abstract:
Some mathematical aspects of the stochastic cosmology are discussed in its relationship to the corresponding ordinary Friedman world models. In particular, it is shown that if the strong and Lorentz energy conditions are known, or the potential function is given, or a stochastic measure is suitable defined then the structure of the phase plane of the Friedman dynamical system is determined.

30. J.Gruszczak, M. Heller
Singularities in stochastically Predictable Universe
Physics Letters A, vol. 106, pp. 13-16 (1984).
[abstract]

Abstract:
Abstract.