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Analysis of the Lorenz Equations for Large r

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In the limit of large r, the Lorenz equations become “almost” conservative. In this limit, one can use the method of averaging (or some equivalent) to obtain a set of two autonomous differential equations for two slowly varying amplitude functions B and D. A stable fixed point of these equations represents the stable periodic solution which is observed at large r. There is an invariant line B = D on which the method breaks down and the averaged equations are no longer valid. In this paper we show how to extend the validity of the analysis by PoincarĂ© mapping B and D across this line. This extended analysis provides (in principl ) a complete recipe for constructing approximate solutions, and shows how a strange invariant set can occur in connection with an essentially analytically constructed two‐dimensional mapping.
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Document Type: Research Article

Affiliations: Massachusetts Institute of Technology

Publication date: June 1, 1984

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