Mason Porter to Carlos Hoyos, Oct 29 2003:
> On Mon, 27 Oct 2003, Carlos Hoyos wrote:
>
> >
> > dy/dt = f(y(t-T1)) + f(y(t-T2)) + g(t) ; T1 and T2 diferent constants.
> >
> > I would like to ask you if there is any elegant way to integrate it since
> > all I can think of by replacing variables is an infinit set of equations
> > or, what I have already done, use interpolation to obtain y(t-T1) and
> > y(t-T2).
>
One of my professors at Cornell, Richard Rand, has spent a lot of time
studying delay equations, which are like pdes in being
infinite-dimensional. One can apply things like the method of averaging
to them to get equations of motion for slow dynamics (which will be
ode's). The equilibria of the resulting odes are then periodic orbits,
and one can study bifurcations, etc.
Moreover, Rand has studied entrainment a lot (which is a name for
mode-locking phenomena that shows up in the engineering literature), so I
think his methods have the potential to be helpful.
Rand also has an online book on nonlinear vibrations that _may_ have a
section on delay equations. I've attached the most recent version that I
have.
Rand is very nice, so if you e-mail him, he'll respond. I would
suggest asking him about where to look in the literature. Also, many
(most?) of his papers are posted on his website, so you may even be able
to find an appropriate reference there. He is in the T & AM department at
Cornell, www.tam.cornell.edu/randdocs/ .
Also, one of my friends did his doctorate under Rand and studied some
delay equations. (Sometimes, the best way to learn something is to
read somebody's Ph.D. thesis.) Attached are my friend's thesis and
a related paper on delay differential equations.
I hope that some of the stuff in there can help with your work on the
delay in your equation. Take a look at chapter 2, for instance, for a
discussion of averaging to get 'slow flow' equations. Methods like that
could perhaps help you.
I would also recommend looking at the references in the thesis and paper
to see if there is anything useful there.
Finally, Jack Hale and Shui-Nee Chow in the math dept know stuff about
delay equations. Their rigorous theory is probably not going to be what
you want, but whether or not it is, they should be able to give you good
ideas for sources to check out.
When I mentioned the delay type stuff in bacterial growth
dynamics, I believe they use something like Mackey-Glass.
I hope this helps.
Mason
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Mason "I was misquoted." Porter
Visiting Assistant Professor, School of Mathematics
Research Associate Member, Center for Nonlinear Science, School of Physics
Georgia Institute of Technology
Homepage: http://www.math.gatech.edu/~mason/index.html, IM: tepid451
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