7. Conclusions

We have given an introduction to many new ideas in non-linear dynamics which have been used to obtain a qualitative and quantitative description of chaotic behaviour in physical systems. This mathematical machinery allows the Einstein equations to be investigated as a dynamical system and the sources of chaotic behaviour therein isolated. The Mixmaster Universe provides a paradigm for chaotic behaviour in the Einstein equations and by a variety of mathematical techniques it has been possible to extend the earlier analyses of Misner and Belinskii et al. to obtain a detailed description of this behaviour. A Poincard return mapping was found for the Mixmaster dynamics. The use of techniques from the metric theory of numbers enables various absolute invariants of the Mixmaster Universe tc be isolated. We then discussed how the existence of Poincar6 mappings might allow general features of the Einstein equations to be discovered and described how time evolution can be incoiporated. Various strong stochastic properties of the Mixmaster system are easily deduced from its return mapping and can also be found from the Hamiltonian formulation of Misner and Ryan. Finally, we have discussed how the discovery of universal behaviour in systems exhibiting a transition to chaos by period-doublinr might esabte the Einstein equations to be investigated in a new way and provide the notion of 'gravitational entropy' with a mathematical basis.

We have shown that new ideas in non-linear dynamics provide a natural description of the most general behaviour so far detected in Einstein's equations. This enables the concept of a chaotic cosmology' to be made rigorous. Finally, we should add, Einstein's equations create a new field enquiry for dynamicists. A field that should prove exceptionally fertile, for the uniqie self-interacting non-linearity of general relativistic systems hints at the presence of chaotic behavixir of unsuspected subtlety.