From: israel@math.ubc.ca (Robert Israel)
Subject: Re: Lorentz attractor definition
Date: 17 Aug 1999 04:38:29 GMT
Newsgroups: sci.math
Keywords: chaotic ODE

In article <7p8k2f$o4k$1@nnrp1.deja.com>,
David Bernier  <bernier@my-deja.com> wrote:
>I think Lorentz's chaotic ODE is a three-variable ODE.

specifically, x' = -sigma x + sigma y, y' = r x - y - x z,
z' = -b z + x y, where sigma, r and b are constants.


>I would suppose that the strange attractor orbit
>X(t) is unique (up to reparameterization).  How is
>it defined? 

An attractor is a closed subset A of the phase space (R^3
in this case) such that 
1) there is a nonempty open subset N of R^3 such that whenever
(x0,y0,z0) is in N, the solution (x(t),y(t),z(t)) with 
initial value (x0,y0,z0) approaches A as t -> infinity.
2) there is no nonempty closed proper subset of A that has this
property.


> How about (x0, y0, z0) on X iff
>X from time 0 onwards comes within epsilon of (x0, y0, z0)
>for arbitrarily large times, no matter the choice of
>positive epsilon?

No.  You didn't specify what the starting point is.  There
can be fixed points, unstable closed orbits, etc, that are
not part of an attractor.

>  Is X(t) defined for all real t?
>If you run X(t) backwards in time, is it bounded?

In the Lorentz case, it's not bounded as t -> -infinity.  Well, certainly 
there are solutions that are not (for x=y=0, you have z' = -bz).
On the other hand, solutions do exist for all time: they can't run
off to infinity in finite time, because (if R = sqrt(x^2+y^2+z^2))
(R^2)' = x x' + y y' + z z' = - sigma x^2 + (sigma + r) x y - y^2 - b z^2
= O(R^2) implies R^2 <= A exp(B |t|) for some A and B.

Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            
Vancouver, BC, Canada V6T 1Z2