Herbert M Sauro wrote:
> I'm not an expert nor a mathematician but could someone explain to me
> how one should go about (in practical terms) contructing a Poincare
> section for a system which does not have a natural periodicity (e.g an
> externally applied periodic forcing function)?
>> 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
line should you use? That depends on your flow. In a nutshell, you
generally want a line passing through the fixed point (equilibrium or
steady state) near which most of the dynamics occurs. For simplicity
(again) suppose that you have a fixed point at (0,0). Either of the
coordinate axes would do. You could for instance watch for crossings of
the x axis. However, you only want crossings in one particular
direction because if the flow crosses from positive to negative values
of y, it must eventually recross in the opposite direction so this
second group of crossings carries no new information. Your program
looks for pairs of points (xa,ya) and (xb,yb) where ya < 0 and yb > 0
(or vice-versa, according to taste). The program then interpolates to
find the value of x at which the axis was crossed, i.e. at which y=0.
The traditional way to treat this data is to pair up successive
values of x at which the y axis was crossed in a specified direction.
Call these doublets (x1,x2) and plot these out. This is sometimes
called a first-return map although there are other names for it as well.
Let's go up to three dimensions now. Your surface of section could
be the xy half-plane. You're now looking for pairs of points at which
za > 0 and zb < 0 and interpolate between them to get the (x,y)
coordinates of the point which crossed the plane. Pair these up into a
four-vector (x1,x2,y1,y2). Frequently, people plot x1 vs x2 (or y1 vs
y2) to obtain a first return map.
Note that Bard Ermentrout's xpp (which runs on Unix systems) can do
this somewhat more painlessly than special purpose programming. xpp is
available by anonymous FTP from ftp.math.pitt.edu/pub/bardware.
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.
Marc R. Roussel
(roussel at henri.chem.uleth.ca)
Department of Chemistry
University of Lethbridge
