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Cosmological models expressible as gradient vector fields. (English) Zbl 0867.34024

The paper deals with the classes of cosmological models for which Einstein’s equations reduce to two-dimensional dynamical systems, studied in the presence of stochastic perturbations using the steady state of the associated Fokker-Planck equations and Zeeman’s notion of \(\varepsilon\)-stability. The classes of such equations which are expressible as dissipative gradient vector fields by means of nonlinear transformations are of interest since their \(\omega\)-limit sets consist only of critical points, and in addition the steady states of their corresponding Fokker-Planck equations can be found analytically. It occurs that in the two-dimensional cases considered in the paper the critical points which do not lie at infinity are not stable, and in the most cases the critical point is a saddle which results in decaying and exponentially expanding directions for the steady state function \(s\), corresponding dynamically to the directions of the stable and unstable manifolds. It has been shown that the stable fixed points at infinity, and the form of the function \(s\) in the neighborhood of these fixed points, can be examined by a careful compactification of the phase plane. In this case, \(s\) remains bounded in all directions, indicating the attracting nature of the fixed points in all directions.

MSC:

34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
83F05 Relativistic cosmology
37-XX Dynamical systems and ergodic theory
34D05 Asymptotic properties of solutions to ordinary differential equations
34F05 Ordinary differential equations and systems with randomness

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