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A description of the Lorenz attractor at high Prandtl number. (English) Zbl 1194.37155

Summary: We use the ideas of matched asymptotic expansions to construct a ‘solution’ of the Lorenz equations in the limit \(r\approx \sigma \rightarrow \infty \). Unlike the case \(r\rightarrow \infty , \sigma \approx 1\) which exhibits limit cycle behaviour, the solutions in the present case can be ‘chaotic’; nevertheless, their structure can be analysed. Our aim is to obtain an analytic approximation of the ‘Lorenz map’ which plots successive maxima of the variable \(Z\). Particularly, we are able to explain the peculiar cusp of this map. and indeed to predict the existence of many such cusps. It is our hope that the methods and ideas presented here may be of use in elucidating the nature of other systems which exhibit chaotic behaviour.

MSC:

37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
76F99 Turbulence
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