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DDE delay differential equations, limit of small delays.


From: Claude Lacoursiere
Subject: DDE delay differential equations, limit of small delays.
Date: Wed, 12 Feb 2014 04:42:46 -0800 (PST)

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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