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We review the asymptotic properties of spatially homogeneous perfect fluid models in general relativity. We then discuss some results concerning scalar field models with an exponential potential both with and without barotropic matter.

Finally, we discuss some isotropic cosmological models derived from the string effective action. However, many aspects of the theory are not yet fully understood theoretically. In this thesis, the general relativistic gravitational field equations are studied using the theory of dynamical systems. Such systems cover a wide variety of interesting phenomena in cosmology and astrophysics.

## Dynamical Systems And Cosmology

For example, special spherically symmetric models yield dynamical systems that are important for understanding gravitational collapse, cosmic censorship, and non-linear density fluctuations in standard cosmology. Various techniques are discussed and developed. Analytic and numerical methods are combined in order to obtain a global picture of the solution space of the systems of the differential equations considered.

These results are then used to extract physical information. The most simple choice gives. The graph of function X t is the upper half right part of an hyperbola. The absciss of its center is —p and its ordinate is 0. The square of the speed of evolution is. Starting from equation Eq.

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It is the limit of the half ellipse seen in the elliptic case. The square of the speed is. So, as we have seen, we have a push back and a push forward. We have. Temperature at point x, at reference instant t, is u t,x. The partial derivative equation of evolution is. We consider models for which the state of the universe, at reference instant t, is given by the so called scalar factor R t. We recognize, according to Eq. Some of them may not be realistic, but due to the uncertainty concerning our conception of the universe and its evolution, they must not be discarded systematically.

For example we may have a universe with no matter and no radiation, at least as an approximation.

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As a first case we add that space has a negative curvature and a negative cosmological constant. Another possibility is the case of a universe with no matter nor radiation as above but with positive curvature and positive cosmological constant. We shall consider some other cases, also presented by Berry, where radiation is negligeable.

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We start with the hypothesis of a flat space and a negative cosmological constant. There are other interesting cases with negligible radiation and null cosmological constant. It is easier to represent the graph of function R t parametrically than explicitely. This new approach will put, metaphorically speaking, emphasis on perception.

The purpose being to propose a modelling of the perception duration. First we consider the linear differential equation. When a t is a mere constant a, it means obviously that a is strictly positive. The sign — has been placed before the integral to make more evident that the positivity of a has this consequence.

This formulation is nor irrealistic and is well adapted to the modellisation of a tachymeter or an accelerometer if we consider mainly the first or the second derivatives the other terms being rather negligible.

## CiteSeerX — Dynamical Systems and Cosmology

Here this operator acts in an instantaneous way, being purely local. The first factor b 0 t , or more simply b t , may be considered positive when it is null there is no attention and so no perception at the considered instant t. For example we may have a convolution.

We must have. Let us come back to the differential equation and its solution Eq. The property of transitivity Eq. In other words we may conclude that the more ancient is a perception the more feeble is its remembrance. If our differential system starts with the null state we have according to Eq. We may interpret x t as the result of the superposition of all the successive perceptions, from t 0 to t, as they are transferred to t by memorization.

It may be compared to the factor of forgetfulness Vogel, or the memory coefficient Allais, We shall consider now differential equation Eq. Taking into account Eq. Of course t is a good parametrisation of time, in the sense that two distinct instants are represented by two different values of t. We consider that function u t gives the evolution of the weight of importance of reference instant t. So we may write simply.

In our model we have an imperfect memorization when a t is not identical to 0. The factor of memorization. The remembrance of past perceptions vanishes with time.