On the stable part of the Lorenz attractor



For a subcritical value of the Rayleigh number the Lorenz attractor has one unstable and two stable stationary states. The asymptotic state of a long time integration may one of the two steady states. A stable steady state will be the asymptotic state if the starting position is in the immediate neighbourhood of the steady state. However, in the region midway between the two stable states the asymptotic state can be found only by a numerical integration. Small changes in the initial state may cause a change in the asymptotic state from one stable steady state to the other. The final asymptotic state is sensitive not only to the small changes in the initial state, but also to small changes in the value of the Rayleigh number. This behavior indicates that at least the border region may be of a fractal nature. In view of the behavior of the time integrations a stochastic model, including only second order statistics, was formulated. This model can be used to investigate the asymptotic states as a function of the magnitude of the uncertainty in the initial state. Several examples will be given. The stochastic model is compared with an exact solution for the steady states given the initial position and the initial variances.

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