More Fractal Geometry/Chaos Theory

[Matthew P Wiener]

Ralph Howard writes:

>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 on the audience.

You are being tediously silly.

>				  Freshman calculus students are often
>quite blown away at the growth rate of exponential functions.

And their opinion does not count.  The technical stand-ins for "extreme
sensitivity" etc, are not understood by consulting with freshman calculus
students.

>							        My
>point, apparently poorly made, is that nonlinear dynamical systems
>have the same sensitivity to initial conditions as linear dynamical
>systems.

But they don't.  The relative error in your example remains incredibly
minuscule.  What is intriguing is when the same input error leads to an
error of the size of a meter in a system that is confined to a meter.

>> 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).

I'll say.

>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.

And it is a totally irrelevant sense.  What matters is the *relative*
error in an actual situation, not your best *general* estimate of the
wrong absolute error.  Nor can you claim too much from looking at just
"the standard condition".  If you have a non-Lipshitz problem, you have
a non-Lipshitz problem, and Mr Gronwall is not going to help you.

>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.

They are more sensitive.  The spatial bounding means the relative
error grows in the chaotic case.  The linear case outraces its
error, so it maintains a comparatively minuscule relative error.
--
-Matthew P Wiener (weemba at sagi.wistar.upenn.edu)