# Delay differential equations in Python

I wrote a very simple and user-friendly method, that I called ddeint, to solve delay differential equations (DDEs) in Python, using the ODE solving capabilities of the Python package Scipy. As usual the code is available at the end of the post :).

## Example 1 : Sine

I start with an example whose exact solution is known so that I can check that the algorithm works as expected. We consider the following equation:

$y(t) = sin(t) \ \ \ for\ \ \ t < 0$

$y'(t) = y(t-3\pi/2) \ \ \ for\ \ \ t \geq 0$

The trick here is that $sin(t-3\pi/2) = cos(t)=sin'(t)$ so the exact solution of this equation is actually the sine function.

from pylab import *

model = lambda Y,t : Y(t - 3*pi/2) # Model
tt = linspace(0,50,10000) # Time start, end, and number of points/steps
g=sin # Expression of Y(t) before the integration interval
yy = ddeint(model,g,tt) # Solving

# PLOTTING
fig,ax=subplots(1)
ax.plot(tt,yy,c='r',label="$y(t)$")
ax.plot(tt,sin(tt),c='b',label="$sin(t)$")
ax.set_ylim(ymax=2) # make room for the legend
ax.legend()
show()


The resulting plot compares our solution (red) with the exact solution (blue). See how our result eventually detaches itself from the actual solution as a consequence of many successive approximations ? As DDEs tend to create chaotic behaviors, you can expect the error to explode very fast. As I am no DDE expert, I would recommend checking for convergence in all cases, i.e. increasing the time resolution and see how it affects the result. Keep in mind that the past values of Y(t) are computed by interpolating the values of Y found at the previous integration points, so the more points you ask for, the more precise your result.

## Example 2 : Delayed negative feedback

You can select the parameters of your model at integration time, like in Scipy’s ODE and odeint. As an example, imagine a product with degradation rate r, and whose production rate is negatively linked to the quantity of this same product at the time (t-d):

$y(t) = 0 \ \ \ for\ \ \ t < 0$

$y'(t) = \dfrac{1}{1+(\dfrac{y(t-d)}{K})^2} -ry(t) \ \ \ for\ \ \ t \geq 0$

We have three parameters that we can choose freely. For K = 0.1, d = 5, r = 1, we obtain oscillations !

from pylab import *

# MODEL, WITH UNKNOWN PARAMETERS
model = lambda Y,t,k,d,r :  1/(1+(Y(t-d)/k)**2) - r*Y(t)

# HISTORY
g = lambda t:0

# SOLVING
tt = linspace(0,50,10000)
yy = ddeint(model,g,tt,fargs=( 0.1 , 5 , 1 )) # K = 0.1, d = 5, r = 1

# PLOTTING
fig,ax=subplots(1)
ax.plot(tt,yy,lw=2)
show()


## Example 3 : Lotka-Volterra system with delay

The variable Y can be a vector, which means that you can solve DDE systems of several variables. Here is a version of the famous Lotka-Volterra two-variables system, where we introduce some delay d. For d=0 we find the solution of a classical Lotka-Volterra system, and for d non-nul, the system undergoes an important amplification:

$\big(x(t), y(t)\big) = (1,2) \ \ \ for\ \ t < 0, \ \ else$

$x'(t) = 0.5x(t)\big(1-y(t-d)\big)\\ y'(t) = -0.5y(t)\big(1-x(t-d)\big)$

from pylab import *

def model(Y,t,d):
x,y = Y(t)
xd,yd = Y(t-d)
return array([0.5*x*(1-yd), -0.5*y*(1-xd)])

g = lambda t : array([1,2])
tt = linspace(2,30,20000)
fig,ax=subplots(1)

for d in [0, 0.2]:
yy = ddeint(model,g,tt,fargs=(d,))
# WE PLOT X AGAINST Y
ax.plot(yy[:,0],yy[:,1],lw=2,label='delay = %.01f'%d)

ax.legend()
show()


## Example 4 : A DDE with varying delay

This time the delay depends on the value of Y(t) !

$y(t) = 1,\ \ \ t \leq 0$

$y'(t) = - y\big(t-3\cos(y(t))^2 \big),\ \ \ t > 0$

from pylab import *
model = lambda Y,t:  -Y(t-3*cos(Y(t))**2)
tt = linspace(0,30,2000)
yy = ddeint(model, lambda t:1, tt)
fig,ax=subplots(1)
ax.plot(tt,yy,lw=2)
show()


## Code

### Explanations

The code is written on top of Scipy’s ‘ode’ (ordinary differential equation) class, which accepts differential equations under the form

model(Y,t) = ” expression of Y'(t) ”

where $Y$, and the output $Y'$, must be Numpy arrays (i.e. vectors).

For our needs, we need the input $Y$ to be a function of time, more precisely a function that can compute $Y(t)$ at any past or present $t$ using the values of $Y$ already computed. We also need $Y(t)$ to return the value of some function $g(t)$ if t is inferior to some time $tc$ that marks the start of the integration.

To this end, I first implemented a class (ddeVar) of variables/functions which can be called at any time $t$: for $t<latex t_c$, it will return the value of $g(t)$, and for $t>tc$, it will look for two already computed values $Y_a$ and $Y_b$ at times $t_a, from which it will deduce $Y(t)$ using a linear interpolation. Scipy offers many other kinds of interpolation, but these will be slower and won't support vectors for $Y$.

Such variables need to be updated every time a new value of $Y(t)$ is computed, so I created a class 'dde' that inherits from Scipy's 'ode' class but overwrites its integration method so that our special function $Y$ is updated after each integration step. Since 'ode' would feed the model with a vector Y (a Numpy array to be precise), which we don't want, we give to the integrator an interface function that takes a Numpy array Y as an argument, but immediately dumps it and calls the model with our special ddeVar variable Y (I hope that was clear ðŸ™‚ ).

### Ok, here you are for the code

You will find the code and all the examples as an IPython notebook HERE (if you are a scientific pythonist and you don’t know about the IPython notebook, you are really missing something !). Just change the extension to .ipynb to be able to open it. In case you just asked for the code:

# REQUIRES PACKAGES Numpy AND Scipy INSTALLED
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
self.tc= 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<=self.tc) 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)
scipy.integrate.ode.__init__(self,f2,jac)
self.set_f_params(None)

def integrate(self, t, step=0, relax=0):

scipy.integrate.ode.integrate(self,t,step,relax)
self.Y.update(self.t,self.y)
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(Y.tc), Y.tc)

def ddeint(func,g,tt,fargs=None):
""" similar to scipy.integrate.odeint. Solves the DDE system
defined by func at the times tt with 'history function' g
and potential additional arguments for the model, fargs
"""

dde_ = dde(func)
dde_.set_initial_value(ddeVar(g,tt[0]))
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)])


## Other implementations

If you need a faster or more reliable implementation, have a look at the packages pyDDE and pydelay, which seem both very serious but are less friendly in their syntax.

# Extract data from graph pictures with Python

If you want to transform a picture of a graph into exploitable data (which is very useful in science if you want to exploit a figure from an article without bothering the authors), here is a minimalistic interface written in python with the following features:

• Data extraction from picture files or from a picture in the clipboard.
• Data extraction from rotated graphs or graphs shown with (moderate) perspective.
• Advanced interface (left-click to select a point, right-click to deselect).
• Stores the points’ coordinates in a python variable and in the clipboard (for use in another application).

You can launch the interface with

points = pic2data()


This will either start a session using the picture from the clipboard, or , if there is none, wait for the clipboard to contain a picture. Alternatively you can use a picture from a file with

points = pic2data('graph.jpeg')


You will then be asked you to place the origin of the graph, as well as the coordinates of this origin (in case it it not (0,0)), and one reference point for each axis X and Y (i.e. points of these axis whose coordinates you know). Then you can select/deselect as many points of the curve as you want, and exit with the middle button.The list of selected points [(x1,y1),(x2,y2),…] is returned.

By default the program will consider that the graph is rectangular and parralel to the edges of the pictures (wich I will call straight in what follows). This will typically be the case for a graph from a scientific article. As a consequence the algorithm will automatically replace the reference point you chose for the X axis in order to put it at the same height as the origin, and it will replace the reference point for Y exactly above the origin. However if the graph on the picture is not straight, like in a photo, use the argument straight=False.

As an example, let us take a photo with a graph, like this one.

Fig. 1: Young Frederic Chopin disguised as Mozart.

As the graph is not straight we will use

points = pic2data('mozart.jpeg', straight = False)


Which gets you to that:

After placing the points and getting their coordinates one can redraw the plot with

from pylab import *
figure()
x,y = zip(*points)
plot(x,y,'o')
show()


And voilÃ  !

Here is the code. Happy curving !

from urlparse import urlparse

import pygtk
import gtk
import tkSimpleDialog

import matplotlib.image as mpimg
import matplotlib.pyplot as plt

import numpy as np

def tellme(s):
print s
plt.title(s,fontsize=16)
plt.draw()

def pic2data(source='clipboard',straight=True):
""" GUI to get data from a XY graph image. Either provide the graph
as a path to an image in 'source' or copy it to the clipboard.
"""

##### GET THE IMAGE

clipboard = gtk.clipboard_get()

if source=='clipboard':

# This chunk tries the text content of the clipboard
# and empties it if it is not a file path

print "Waiting for an image in the clipboard..."
while not ( clipboard.wait_is_uris_available()
or clipboard.wait_is_image_available()):
pass

if clipboard.wait_is_uris_available(): # it's a path to a file !

source = clipboard.wait_for_uris()[0]
source = urlparse(source).path
return pic2data(source)

image = clipboard.wait_for_image().get_pixels_array()
origin = 'upper'

else: # source is a path to a file !

origin = 'lower'

###### DISPLAY THE IMAGE

plt.ion() # interactive mode !
fig, ax = plt.subplots(1)
imgplot = ax.imshow(image, origin=origin)
fig.canvas.draw()
plt.draw()

##### PROMPT THE AXES

def promptPoint(text=None):

if text is not None: tellme(text)
return  np.array(plt.ginput(1,timeout=-1)[0])

initialvalue=initialvalue)

origin = promptPoint('Place the origin')

Xref =  promptPoint('Place the X reference')

Yref =  promptPoint('Place the Y reference')

if straight :

Xref[1] = origin[1]
Yref[0] = origin[0]

##### PROMPT THE POINTS

selected_points = []

print "Right-click or press 's' to select"
print "Left-click or press 'del' to deselect"
print "Middle-click or press 'Enter' to confirm"
print "Note that the keyboard may not work."

selected_points = plt.ginput(-1,timeout=-1)

##### RETURN THE POINTS COORDINATES

#~ selected_points.sort() # sorts the points in increasing x order

# compute the coordinates of the points in the user-defined system

OXref = Xref - origin
OYref = Yref - origin
xScale =  (Xref_value - origin_value[0]) / np.linalg.norm(OXref)
yScale =  (Yref_value - origin_value[1]) / np.linalg.norm(OYref)

ux = OXref / np.linalg.norm(OXref)
uy = OYref / np.linalg.norm(OYref)

result = [(ux.dot(pt - origin) * xScale + origin_value[0],
uy.dot(pt - origin) * yScale + origin_value[1])
for pt in selected_points ]

# copy the result to the clipboard

clipboard.set_text('[' + '\n'.join([str(p) for p in result]) + ']')

clipboard.store() # makes the data available to other applications

plt.ioff()

return result


# Animate your 3D plots with Python’s Matplotlib

When you have a complicated 3D plot to show in a video or slideshow, it can be nice to animate it:

I obtained this surface with

import matplotlib.pyplot as plt
from matplotlib import cm
from mpl_toolkits.mplot3d import axes3d

fig = plt.figure()
X, Y, Z = axes3d.get_test_data(0.05)
s = ax.plot_surface(X, Y, Z, cmap=cm.jet)
plt.axis('off') # remove axes for visual appeal


To animate it I created the function rotanimate that you can use like that:

import numpy as np
angles = np.linspace(0,360,21)[:-1] # A list of 20 angles between 0 and 360

# create an animated gif (20ms between frames)
rotanimate(ax, angles,'movie.gif',delay=20)

# create a movie with 10 frames per seconds and 'quality' 2000
rotanimate(ax, angles,'movie.mp4',fps=10,bitrate=2000)

# create an ogv movie
rotanimate(ax, angles, 'movie.ogv',fps=10)


Here is the source-code:

import matplotlib.pyplot as plt
from matplotlib import cm
from mpl_toolkits.mplot3d import axes3d
import os, sys
import numpy as np

##### TO CREATE A SERIES OF PICTURES

def make_views(ax,angles,elevation=None, width=4, height = 3,
prefix='tmprot_',**kwargs):
"""
Makes jpeg pictures of the given 3d ax, with different angles.
Args:
ax (3D axis): te ax
angles (list): the list of angles (in degree) under which to
take the picture.
width,height (float): size, in inches, of the output images.
prefix (str): prefix for the files created.

Returns: the list of files created (for later removal)
"""

files = []
ax.figure.set_size_inches(width,height)

for i,angle in enumerate(angles):

ax.view_init(elev = elevation, azim=angle)
fname = '%s%03d.jpeg'%(prefix,i)
ax.figure.savefig(fname)
files.append(fname)

return files

##### TO TRANSFORM THE SERIES OF PICTURE INTO AN ANIMATION

def make_movie(files,output, fps=10,bitrate=1800,**kwargs):
"""
Uses mencoder, produces a .mp4/.ogv/... movie from a list of
picture files.
"""

output_name, output_ext = os.path.splitext(output)
command = { '.mp4' : 'mencoder "mf://%s" -mf fps=%d -o %s.mp4 -ovc lavc\
-lavcopts vcodec=msmpeg4v2:vbitrate=%d'
%(",".join(files),fps,output_name,bitrate)}

command['.ogv'] = command['.mp4'] + '; ffmpeg -i %s.mp4 -r %d %s'%(output_name,fps,output)

print command[output_ext]
output_ext = os.path.splitext(output)[1]
os.system(command[output_ext])

def make_gif(files,output,delay=100, repeat=True,**kwargs):
"""
Uses imageMagick to produce an animated .gif from a list of
picture files.
"""

loop = -1 if repeat else 0
os.system('convert -delay %d -loop %d %s %s'
%(delay,loop," ".join(files),output))

def make_strip(files,output,**kwargs):
"""
Uses imageMagick to produce a .jpeg strip from a list of
picture files.
"""

os.system('montage -tile 1x -geometry +0+0 %s %s'%(" ".join(files),output))

##### MAIN FUNCTION

def rotanimate(ax, angles, output, **kwargs):
"""
Produces an animation (.mp4,.ogv,.gif,.jpeg,.png) from a 3D plot on
a 3D ax

Args:
ax (3D axis): the ax containing the plot of interest
angles (list): the list of angles (in degree) under which to
show the plot.
output : name of the output file. The extension determines the
kind of animation used.
**kwargs:
- width : in inches
- heigth: in inches
- framerate : frames per second
- delay : delay between frames in milliseconds
- repeat : True or False (.gif only)
"""

output_ext = os.path.splitext(output)[1]

files = make_views(ax,angles, **kwargs)

D = { '.mp4' : make_movie,
'.ogv' : make_movie,
'.gif': make_gif ,
'.jpeg': make_strip,
'.png':make_strip}

D[output_ext](files,output,**kwargs)

for f in files:
os.remove(f)

##### EXAMPLE

if __name__ == '__main__':

fig = plt.figure()
X, Y, Z = axes3d.get_test_data(0.05)
s = ax.plot_surface(X, Y, Z, cmap=cm.jet)
plt.axis('off') # remove axes for visual appeal

angles = np.linspace(0,360,21)[:-1] # Take 20 angles between 0 and 360

# create an animated gif (20ms between frames)
rotanimate(ax, angles,'movie.gif',delay=20)

# create a movie with 10 frames per seconds and 'quality' 2000
rotanimate(ax, angles,'movie.mp4',fps=10,bitrate=2000)

# create an ogv movie
rotanimate(ax, angles, 'movie.ogv',fps=10)


# A GUI for the exploration of functions with Python / Matplotlib

Sliders can be a great tools for lazy engineers and modelists who want to play around with the parameters of a program without recompiling it a thousand times.
Most scientific languages ( python-pylab, matlab, scilab…) support some basic sliders methods but it can be quite long to create a GUI. In this blog I present a light, user-friendly function I implemented for my everyday work.

## Example with a numerical function

def volume(x,y,z):
""" Volume of a box with width x, heigth y, and depth z """
return x*y*z

intervals = [ { 'label' :  'width',  'valmin': 1 , 'valmax': 5 },
{ 'label' :  'height',  'valmin': 1 , 'valmax': 5 },
{ 'label' :  'depth',  'valmin': 1 , 'valmax': 5 } ]

inputExplorer(volume,intervals)



The new value of volume is automatically computed each time you move the sliders. It is also computed if you press Enter (which can be useful to get several values, when your function is non-deterministic). In case you are studying a function that is long to evaluate, you can decide that it will only be evaluated when Enter is pressed, by adding “wait_for_enter = True” to the arguments list.

## Example with a graphical function

You can also decide to use a function that doesn’t return any value, but updates a plot. Let us illustrate this with the Lotka-Volterra model, in which wolves eat rabbits and die, leading to periodical fluctuations of both populations. From wikipedia:

$\frac{dx}{dt} = x(\alpha - \beta y)$
$\frac{dy}{dt} = - y(\gamma - \delta x)$

This can be simulated easily in python with scipy’s function odeint :

import numpy as np
from scipy.integrate import odeint

# model

def model(state,t, a,b,c,d):
x,y = state
return [ x*(a-b*y) , -y*(c - d*x) ] # dx/dt, dy/dt

# vector of times

t_vec = np.linspace(0,10,500) # 500 time points between 0 and 10

# initial conditions

x0 = 0.5
y0 = 1

# parameters

a = 1
b = 2
c = 1
d = 3

# simulate and return the values at each time t in t_vec

result = odeint(model, [x0,y0], ts, args = (a,b,c,d) )

# plot

import matplotlib.pyplot as plt
plt.plot(ts,result)
plt.show()


However, trying by hand several values of x0, y0, a, b, c, d, can be fastidious. So let us wrap the solving and the plotting in a function and feed it to inputExplorer. Let me rewrite the whole code from scratch :

import matplotlib.pyplot as plt
import numpy as np
from scipy.integrate import odeint

def model(state,t, a,b,c,d):
x,y = state
return [ x*(a-b*y) , -y*(c - d*x) ]

ts = np.linspace(0,10,500)

fig,ax = plt.subplots(1)

def plotDynamics(x0,y0,a,b,c,d):
ax.clear()
ax.plot(ts, odeint(model, [x0,y0], ts, args = (a,b,c,d)) )
fig.canvas.draw()

sliders = [ { 'label' :  label,  'valmin': 1 , 'valmax': 5 }
for label in [ 'x0','y0','a','b','c','d' ] ]

inputExplorer(plotDynamics,sliders)



## Code

Enjoy !

import matplotlib.pyplot as plt
from matplotlib.widgets import Slider, Button

def inputExplorer(f, sliders_properties, wait_for_validation = False):
""" A light GUI to manually explore and tune the outputs of
a function.
slider_properties is a list of dicts (arguments for Slider )
whose keys are in ( label, valmin, valmax, valinit=0.5,
valfmt='%1.2f', closedmin=True, closedmax=True, slidermin=None,
slidermax=None, dragging=True)

def volume(x,y,z):
return x*y*z

intervals = [ { 'label' :  'width',  'valmin': 1 , 'valmax': 5 },
{ 'label' :  'height',  'valmin': 1 , 'valmax': 5 },
{ 'label' :  'depth',  'valmin': 1 , 'valmax': 5 } ]
inputExplorer(volume,intervals)
"""

nVars = len(sliders_properties)
slider_width = 1.0/nVars
print slider_width

# CREATE THE CANVAS

figure,ax = plt.subplots(1)
figure.canvas.set_window_title( "Inputs for '%s'"%(f.func_name) )

# choose an appropriate height

width,height = figure.get_size_inches()
height = min(0.5*nVars,8)
figure.set_size_inches(width,height,forward = True)

# hide the axis
ax.set_frame_on(False)
ax.get_xaxis().set_visible(False)
ax.get_yaxis().set_visible(False)

# CREATE THE SLIDERS

sliders = []

for i, properties in enumerate(sliders_properties):

ax = plt.axes([0.1 , 0.95-0.9*(i+1)*slider_width,
0.8 , 0.8* slider_width])
sliders.append( Slider(ax=ax, **properties) )

# CREATE THE CALLBACK FUNCTIONS

def on_changed(event) :

res = f(*(s.val for s in sliders))

if res is not None:

print res

def on_key_press(event):

if event.key is 'enter':

on_changed(event)

figure.canvas.mpl_connect('key_press_event', on_key_press)

# AUTOMATIC UPDATE ?

if not wait_for_validation:

for s in sliders :

s.on_changed(on_changed)

# DISPLAY THE SLIDERS

plt.show()