More Fractal Geometry/Chaos Theory

[Ralph Howard]

Matthew P Wiener writes about my example of "extreme sensitivity to
initial conditions" in a linear dynamical system:

> > 
> > If t is time in seconds and y(t) is measured in meters then and we
> > run the dynamic system for a minute (t=60) then to get accuracy of 1
> > meter we need to measure the initial condition to within 8.75 x
> > 10^(-27)meters which is much much much smaller than not only an atom
> > (about 10^(-10)meters), but even the nucleus of an atom (about
> > 10^(-15)meters).

> Accuracy of 1 meter in this case is *extremely* accurate, since the y
> blows up exponentially.  Roughly at the scale of a nucleus on meter
> sized objects.  Not a surprise.

This depends a on the audience.  Freshman calculus students are often
quite blown away at the growth rate of exponential functions.  My
point, apparently poorly made, is that nonlinear dynamical systems
have the same sensitivity to initial conditions as linear dynamical
systems.  I expressed this as (with your comments) as:

> > Moreover there is a result that under the standard restrictions on
> > the parameters of a dynamical system which insure the system will
> > have solutions existing for all time the one with the greatest
> > sensitivity to initial conditions is (and you guessed it from the
> > pedantic tone of my voice) the linear system with the given
> > parameters.  Thus in some sense the linear equations are the MOST
> > sensitive to initial conditions.
> 
> Your comments are complete nonsense.  "The standard restrictions on the
> parameters of a dynamical system"?  What the heck are you talking about?


In reading this over I agree I have written non-sense (doing one of
the things that puts me off about bad popular science writing, using
jargon that has not been explained).  Here is the correct statement
(and I see no way out of using formulas in this case for scalar
dynamical systems, but the case of systems only differs by notation).

*** Aside for Matthew and other math lovers  ***

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).  That is the "worst case"
in the Gronwall inequality is linear equation with the parameter C.
This is the sense that I meant that linear equations are most
sensitive to initial conditions.  

To anticipate a possible objection I note that many (and probably
most) of the standard examples of dynamical systems that are generally
agreed to be chaotic, for example the system generating the Lorenz
mask, satisfy this condition near the mask and whence it is less
sensitive to initial conditions than a linear system with the same
parameter C.  Therefore the chaotic behavior is not due to the
sensitivity to the initial conditions (as there is a linear system
which is not chaotic but is even more sensitive to initial
conditions).  I think that we do agree that chaos is certainly a
fundamentally nonlinear phenomenon.

More of our earlier exchange:

> 
> >that have fixated on the sensitivity to initial conditions as a
> >fundamentally nonlinear phenomenon which is just false.
> 
> No, it is a fundamentally nonlinear phenomenon.

I hope the above will make you reconsider this.  


*** End of Aside  ***


Sanjay Krishnaswamy writes about my example:

> This misses the point entirely.  For the linear system -- by
> definition! -- _arbitrarily close_ initial conditions will produce
> _arbitrarily close_ results.  (Or, to be formal, invoke the "for eny
> epsilon greater than zero there exists a delta such that f(x+delta)
> - f(x) < epsilon" shtick -- proof left to reader!)

As Andrew points out this is the definition of continuity not
linearity.  (I have read your later post which does clear this up a
bit.) Moreover the aside above gives explicit formulas for finding
delta in terms of epsilon so for nonlinear dynamical systems there is
continuous dependence on initial condition just as there is for linear
dynamical systems.

Farther it is almost an axiom in the sciences that a dynamical system
that does not have continuous dependence on initial conditions is a
very bad model.  The reason being that what the dependence is not
continuous then it is not possible specify initial conditions with
enough precision to get useful answers.

Ralph
-- 
Ralph Howard                    Phone:  (803) 777-2913 
Department of Mathematics       Fax:    (803) 777-3783 
University of South Carolina    e-mail: howard at math.sc.edu
Columbia, SC 29208 USA          http://www.math.sc.edu/~howard/