115. 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. |
116. 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. |
117. 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. |
118. 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. |
119. Marek Szydłowski, Michael Heller, Zdzisław Golda Structural Stability Properties of Friedman Cosmology Gen. Rel. Grav., vol. 16, pp. 877-890 (1984). [abstract] [journal] |
Abstract: A dynamical system with Robertson-Walker symmetries and the equation of the state $p = \gamma\epsilon$, $0 \leq\gamma\leq 1$, 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 ($x = R^D$, $y = \dot{x}$). |
120. Z. Golda, M. Heller and M. Szydłowski Structurally Stable Approximations to Friedmann—Lemaître World Models Astrophys. Space Sci., vol. 90, pp. 313-326 (1983). [abstract] [journal] |
Abstract: Friedmann{--}Lema\^{\i}tre cosmology is briefly reviewed in terms of dynamical systems. It is demonstrated that in certain cases bulk viscosity dissipation structurally stabilizes Friedmann{--}Lema\^{\i}tre solutions. It turns out that, for $\Lambda = 0$, there are structurally stable solutions if $\zeta \sim \varepsilon^{1\slash2}$, where $\zeta$ is the bulk viscosity coefficient. For $\Lambda \neq 0$, structurally stable solutions are essentially those with $\zeta = \mbox{const}$. The role of structural stability in physics and cosmology is shortly discussed. |