The Lorenz Model - Demo 7

Return Map

Lorenz noted that the maximum values of the Z variable obtained on successive orbits around one or other fixed point seemed to have some predictability. He therefore plotted a return map of successive maximum values of this variable i.e. plot the (n+1)th value of Zmax against the nth value of Zmax

The return map (that is filled in after some long iteration time) has the appearance of a function (i.e. is single valued), and leads us later to study "one dimensional maps". (Again, looking on a very fine scale we would find that the points do not strictly define a function - there is some slight "fuzziness" to the line.) A great deal of information about the Lorenz dynamics can be intuitively understood from the return map. For example the intersection of the "function" with the diagonal line corresponds to a fixed point; the fact that the magnitude of the slope of the function is greater than unity shows that the fixed point is unstable. Indeed we can associate the chaotic motion with the fact that the magnitude of the slope is greater than unity everywhere. This leads us to the idea of the "sensitivity to initial consitions".

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Last modified Thursday, November 18, 1999
Michael Cross