More Fractal Geometry/Chaos Theory

[Joaquin Stick]

On Tue, 29 Jul 1997, Ralph Howard wrote:

> Consider the dynamical system
>  
>     y'(t)=f(t,y(t))
> 
> For a general f(t,y) this need not have solutions defined for all
> time.  The standard condition that gives long term existence is that
> f(t,y) satisfy the inequality
> 
>    |f(t,y)-f(t,z)| =< C|y-z|
> 
> for some constant C (and C is the "parameter" I was misnaming).  If
> this holds a basic result is that if y(t) and z(t) are solutions to
> the system then a basic result is the Gronwall's Inequality:
> 
>    |y(t)-z(t)| =< |y(0)-z(0)|exp(Ct)     for   t > 0
> 
> which gives an explicit estimate difference of two solutions in terms
> of the initial conditions y(0) and z(0).  Note that if y(t) and z(t) are
> solutions to the linear dynamical system y'(t)=Cy(t) then we have the
> equality  |y(t)-z(t)| = |y(0)-z(0)|exp(Ct).  [...]

I think something is wrong with your word-processor. I keep trying to read
the messages in this thread and all that I see are a bunch of xs, ys and
zs that are mingled in with what looks like machine language code. 

Sigh...I wish I knew more about math.
No, wait, I take that back. All this is giving me a headache. I'm going
back to reading some fluffy stuff like Recherches du Temps Perdu in
Korean.

I wish there was some way to translate this thread into somewhat more
comprehenible terms for those of us who find the concepts of such stuff
interesting but who break out in hives at the mere mention of asymptotes
and logarithms (hell, I always thought they were the beats that natives
made on tree trunks in _Heart of Darkness_). 

Anyway, "carry on at infinite length, my dear"

D. Alfred Fledermaus