We discuss universal statistical patterns in the chaotic dynamics of closed inflationary cosmologies, associated with the presence of a saddle-center critical point in the phase space of the models. In the regime of high nonintegrability, the distribution is universal in the sense that it is scale invariant, independent of the parameters of the model and independent of the particular system where the saddle-center is present. The consequences for the physics of the early stages of inflation are discussed.
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