Probability density of the Lorenz model

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Abstract
The three-dimensional flow of the Lorenz model on its strange attractor is approximated by a two-dimensional flow with a branch curve with the use of the approximation of the Lorenz attractor by invariant two-dimensional manifolds obtained in earlier work. The Poincaré map on the branch curve and the associated invariant measure are determined. The probability density generated by the flow on the invariant manifolds in the steady state is related to and computed from the invariant measure on the branch curve. In a second part of this paper it is shown how the same probability density arises in the Lorenz model subject to stochastic forcing as a self-consistent approximation for the case of very small but finite noise intensity. The distribution transverse to the attractor is determined in the same approximation.

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