Time Scales

This is going to be a very, very mathsy post though hopefully not too hard to understand for anyone who's done "proper" first year maths. I've screwed around with notation and am pretty ignorant about this sort of stuff, so knowing too much maths might be more of a problem :)

I just finished attending the Australian Mathematical Society Annual Conference. Becuase it's all of maths there were many talks on areas I know nothing about, and when I felt like stretching my brain would go to talks on Logic (which was fascinating and made me think I may have chosen the wrong area to study) or "Unstable non-corpuscular nodes of a scalar quantum field in an expanding accelerating universe" (which utterly broke my brain and made me glad I changed out of Physics). Anyway, I went to one Applied talk (Dynamic Equations on Time Scales) and I thought the ideas were so cool, and simple, I'm going to natter about them here :)

This is a differential equation:
x'(t)=3x(t)+t^2, x(0)=0

This is a difference equation:
x(n+1)-x(n)=3x(n)+n^2, x(0)=0
In a sense these equations are "equivalent", but they are not the same.

There is a large area of mathematics devoted to solving differential equations, but as far as I can tell the way difference equations are generally solved is by treating them as if they were differential equations and hoping for the best. The problem being that not only do the solutions differ but there are equations which, say, have infinitely many solutions in the differential case and none in the difference case. For this reason we (by which I mean the guy giving the talk) define dynamic equations.

In this sort of equation (which covers both difference and differential equations) you have:

• A time scale T. This is a closed subset of the real numbers, usually just all the real numbers or the positive integers.
• A function x(t) defined over T.
• For any t in T you define s(t) to be the "next" point, ie in the real numbers s(t)=t and in the integers s(t)=t+1.
• A "derivative" x*(t) where
  • if s(t)=t (ie in a differential equation) then x*(t) is just the derivative x'(t).
  • if s(t)>t (ie in a difference equation) then
    x*(t)=[ x(s(t))-x(t) ]/[ s(t)-t ]. Note that if s(t)=t+1 you just get x*(t)=x(t+1)-x(t).

A simple dynamic equation would then be
x*(t)=3x(t)+t^2, x(0)=0.
If you substitute in s(t)=t or s(t)=t+1 you get the equations we had to begin with. Which all seems a bit pointless except you can define things like the product rule to work for x*(t) regardless of T (you just get some funny [ s(t)-t ] terms which vanish when T is the real numbers) and so come up with results which work for any dynamic equation. Also, and I think this is super cool, you don't have to have a boring old time scale like the integers. Pretty much ANY closed subset of the real numbers will work, including things like

• A collection of continuous intervals ie T={ t: 1<=t<=2 or 4<=t<=5} so you have s(t)=t for most t, but s(2)=4.
• Bounded infinite sequences like {0, 1/2, 3/4, 5/6, ...} (plus the number 1 so it has an upper boundary)
• Crazy sets like the cantor set. Can you imagine differentiating a function which is only defined on the Cantor set? Craaazy!

If you got this far and aren't bored out of your skull, you can read the first chapter of a book on this stuff for free. Also I should mention that the "sort of derivative" x* is actually denoted by x with a superscript delta, but that was too hard to type :)