In article <334EA831.41C6 at henri.chem.uleth.ca>, Marc Roussel
<roussel at henri.chem.uleth.ca> writes
>Herbert M Sauro wrote:
>>>> I have read that one has to put up a n-1 dimensional hyperplane across
>> the flows and then identify the places on the plane where trajectories
>> cross. How does one do this in a practical sense?
>> Let's start with a simple case, namely a Poincare section for a
>two-dimensional system. In this case, your hyperplane is a line. Which
snip, clear stuff.
> If this isn't altogether clear, try drawing yourself some diagrams
>as you read along. If that still doesn't do it for you, send me email
>and I'll clarify any fuzzy points.
Many thanks. What about plotting the current value of a state variable
against it previous value, i.e. xn vs xn+1. If I use a fixed step
integrator shouldn't it give a rough idea of the structure of the
section? I guess the trouble is not all trajectories take a step-size
time to return to the same crossing point. I've seen these plots done
but I've not seen a justification for them. The system I have in mind
happens to be 4 dimensional and using a fixed step integrator and
plotting xn vs. xn+1 I do get a reasonable picture of what looks like a
fish (!). I've read that one can generate a 3 dimensional section first
and process the data further to reduce the section to 2 dimensional but
it it would appear that you need a *lot* of data to do this effectively.
I've also read that Lorenz generated a section by plotting the heights
of the peaks (peakn vs peakn+1), is this reasonable to do (in his day it
obviousy was coz' nobody had done one before - or had poincare done
some?).
> Marc R. Roussel
> (roussel at henri.chem.uleth.ca)
> Department of Chemistry
> University of Lethbridge
Herbert
--
Herbert M Sauro
email: HSauro at fssc.demon.co.uk
Telephone: 01974 282428
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