Re: DDE delay differential equations, limit of small delays.

[Kai Torben Ohlhus]

On Wed, Feb 12, 2014 at 1:42 PM, Claude Lacoursiere <address@hidden> wrote:

I'm investigating the effect of delays on DDEs and though I have some
analytical results, I'd be interesting to test everything.   The problem I
have is that as I let the "lags"  approach 0, I don't get the behavior
needed.  However, if I keep the lag to 1 and rescale time and the dde etc.,
then I get the correct result.  In principle, that's the same number of
steps if there is a small delay or rescaling of time.  However, I get very
different results when using a very small delay or a corresponding time
dilation.

Is there a fundamental limitation in ode45d, or all the other integrators
for DDEs for that matter?  If so, does anyone know when the delay is too
small for the numerical method?

In the script below, I get sensible results up to tau = 0.2.   Below that,
the result of using tau as a lag compared to that using a rescaling are
completely different.

## this should converge to simple exponential decay as lag approaches 0

tau =  0.2;   ## for tau bigger than 0.2, everything makes sense.  At tau  =
0.1, garbage is produced.
lag =  tau;
mu = -1;
dexp = @(vt, vy, vz)  mu * vz ; # as simple as it gets: exponential decay

slot = [0, 10];
x0 = 10;                        # initial condition

vopt = odeset ("NormControl", "on", "RelTol", 1e-6, "MaxStep", 0.5,
"InitialStep", 0.1, "AbsTol", 1E-6);

vsol = ode54d (dexp, slot, x0, lag, x0, vopt);

##rescale everything
mu = tau*mu;
lag = 1;
slot = slot / tau;

dexp = @(vt, vy, vz)  mu * vz ; # as simple as it gets

vsol0 = ode54d (dexp, slot, x0, lag, x0, vopt);

plot (vsol.x, vsol.y, vsol0.x*tau, vsol0.y)



Any comment welcome.

Regards,
Claude



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Hello Claude,

what results did you expect and what did you get from your example code? Maybe check the solver used. Your code uses "ode54d" and in the text above you ask about "limitation in ode45d".

See http://octave.sourceforge.net/odepkg/overview.html

For both solvers I got a solution like shown in the attached picture. Please give more information about your exact problem or the theoretical background of your model you are using and which version of octave and odepkg you are using. Otherwise I think hardly anyone can give you any helpful advice.

Best,

Kai

plot.png
Description: PNG image