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A simple Delay Differential Equation solver written in Python, using the solving capabilities of the Scipy package.
import numpy as np
import scipy.integrate
import scipy.interpolate
class ddeVar:
""" special function-like variables for the integration of DDEs """
def __init__(self,g,tc=0):
""" g(t) = expression of Y(t) for t<tc """
self.g = g tc
# We must fill the interpolator with 2 points minimum
self.itpr = scipy.interpolate.interp1d(
np.array([tc-1,tc]), # X
np.array([self.g(tc),self.g(tc)]).T, # Y
kind='linear', bounds_error=False,
fill_value = self.g(tc))
def update(self,t,Y):
""" Add one new (ti,yi) to the interpolator """
self.itpr.x = np.hstack([self.itpr.x, [t]])
Y2 = Y if (Y.size==1) else np.array([Y]).T
self.itpr.y = np.hstack([self.itpr.y, Y2])
self.itpr.fill_value = Y
def __call__(self,t=0):
""" Y(t) will return the instance's value at time t """
return (self.g(t) if (t< else self.itpr(t))
class dde(scipy.integrate.ode):
""" Overwrites a few functions of scipy.integrate.ode"""
def __init__(self,f,jac=None):
def f2(t,y,args):
return f(self.Y,t,*args)
def integrate(self, t, step=0, relax=0):
return self.y
def set_initial_value(self,Y):
self.Y = Y #!!! Y will be modified during integration
scipy.integrate.ode.set_initial_value(self, Y(,
def ddeint(func,g,tt,fargs=None):
Similar to scipy.integrate.odeint. Solves a Delay differential
Equation system (DDE) defined by ``func`` with history function ``g``
and potential additional arguments for the model, ``fargs``.
Returns the values of the solution at the times given by the array ``tt``.
We will solve the delayed Lotka-Volterra system defined as
For t < 0:
x(t) = 1+t
y(t) = 2-t
For t > 0:
dx/dt = 0.5* ( 1- y(t-d) )
dy/dt = -0.5* ( 1- x(t-d) )
Note that here the delay ``d`` is a tunable parameter of the model.
import numpy as np
def model(XY,t,d):
x, y = XY(t)
xd, yd = XY(t-d)
return np.array([0.5*x*(1-yd), -0.5*y*(1-xd)])
g = lambda t : np.array([1+t,2-t]) # 'history' at t<0
tt = np.linspace(0,30,20000) # times for integration
d = 0.5 # set parameter d
yy = ddeint(model,g,tt,fargs=(d,)) # solve the DDE !
dde_ = dde(func)
dde_.set_f_params(fargs if fargs else [])
return np.array([g(tt[0])]+[dde_.integrate(dde_.t + dt)
for dt in np.diff(tt)])

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@weiszr weiszr commented Dec 9, 2014

I have been trying to play with your solver, but I get the following error: index 2 is out of bounds for axis 0 with size 2. The full error is copied below.

Do you have any idea what the problem is?



Full error:
Traceback (most recent call last):
File "", line 13, in
yy = ddeint(model,g,tt,fargs=( 0.1 , 5 , 1 )) # K = 0.1, d = 5, r = 1
File "/Users/weiszr/Dropbox/apps/Computable/", line 97, in ddeint
for dt in np.diff(tt)])
File "/Users/weiszr/Dropbox/apps/Computable/", line 46, in integrate
File "/Users/weiszr/Library/Enthought/Canopy_64bit/User/lib/python2.7/site-packages/scipy/integrate/", line 388, in integrate
self.f_params, self.jac_params)
File "/Users/weiszr/Library/Enthought/Canopy_64bit/User/lib/python2.7/site-packages/scipy/integrate/", line 737, in run
File "/Users/weiszr/Dropbox/apps/Computable/", line 40, in f2
return f(self.Y,t,_args)
File "", line 6, in
model = lambda Y,t,k,d,r : 1/(1+(Y(t-d)/k)__2) - r_Y(t)
File "/Users/weiszr/Dropbox/apps/Computable/", line 32, in call
return (self.g(t) if (t< else self.itpr(t))
File "/Users/weiszr/Library/Enthought/Canopy_64bit/User/lib/python2.7/site-packages/scipy/interpolate/", line 79, in call
y = self._evaluate(x)
File "/Users/weiszr/Library/Enthought/Canopy_64bit/User/lib/python2.7/site-packages/scipy/interpolate/", line 478, in _evaluate
y_new = self._call(self, x_new)
File "/Users/weiszr/Library/Enthought/Canopy_64bit/User/lib/python2.7/site-packages/scipy/interpolate/", line 441, in _call_linear
y_hi = self._y[hi]
IndexError: index 2 is out of bounds for axis 0 with size 2


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@fkemeth fkemeth commented Dec 3, 2015

Same error here.
Did anyone solve this issue?
Best and Thx,


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@petehellyer petehellyer commented Mar 7, 2016

I also have the same problem, has anyone been able to solve?


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@julianeagu julianeagu commented Oct 31, 2016

Following the solution listed here and here, one needs to add

self.itpr._y = self.itpr._reshape_yi(self.itpr.y)

after line 28. This solved the issue for me.


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@Ziaeemehr Ziaeemehr commented Nov 15, 2016

I compared the result of ddeint and pydelay and here is the code:

""" Reproduces the sine function using a DDE """
from pylab import *
from ddeint import ddeint
from pydelay import dde23
from timeit import default_timer as timer
from numpy import *

start = timer()
model = lambda Y,t : Y(t - 3*pi/2) # Model
tt = linspace(0,50,30000) # Time start, time end, nb of pts/steps
g = sin # Expression of Y(t) before the integration interval
yy = ddeint(model,g,tt) # Solving

end = timer()
print "elapsed time for ddeint : %.4f seconds" % (end - start)


start = timer()
par = {'tau': 1.5*np.pi}
eqns = {'x': 'sin(t-tau)'}
dde = dde23(eqns = eqns, params = par )
dde.set_sim_params(tfinal=50, dtmax=0.1)
histfunc = {
    'y': lambda t : sin(t)
sol = dde.sample(0,50,2e-3)
y1 = sol['x']
t1 = sol['t']
end = timer()
print "elapsed time for pydelay : %.4f seconds" % (end - start)



And here the results:

elapsed time for ddeint : 6.5165 seconds
elapsed time for pydelay : 0.0186 seconds

I should also mention that the resulting curve from pydelay was very close to the analytic solution and ddeint digressed.

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