Scipy-based delay differential equation (DDE) solver. See the docstrings and examples for more infos.
This example is meant to illustrate the solver capabilities of this module as well as how this module can be invoked and what needs to be provided.
# We solve the following system:# Y(t) = 1 for t < 0# dY/dt = -Y(t - 3cos(t)**2) for t > 0frompylabimportcos, linspace, subplotsfromddesolverimportsolve_ddedefmodel(Y, t):
return-Y(t-3*cos(Y(t)) **2)
defvalues_before_zero(t):
return1tt=linspace(0, 30, 2000)
yy=solve_dde(model, values_before_zero, tt)
fig, ax=subplots(1, figsize=(4, 4))
ax.plot(tt, yy)
ax.figure.savefig("variable_delay.jpeg")The following two outputs are that by this module (left) and ddeint by Zulko (right).
This example solves a delayed Lotka–Volterra system with different delays to illustrate how to retain control over parameters of the model system and how delay can affect the results of the solver.
""" We 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) ) The delay ``d`` is a tunable parameter of the model and will be either d=0.0 or d=0.2"""frompylabimportarray, linspace, subplotsfromddesolverimportsolve_ddedefmodel(Y, t, d):
x, y=Y(t)
xd, yd=Y(t-d)
returnarray([0.5*x* (1-yd), -0.5*y* (1-xd)])
g=lambdat: array([1, 2])
tt=linspace(2, 30, 20000)
fig, ax=subplots(1, figsize=(4, 4))
# The parameter d is configurablefordin [0, 0.2]:
print("Computing for d=%.02f"%d)
yy=solve_dde(model, g, tt, fargs=(d,))
# WE PLOT X AGAINST Yax.plot(yy[:, 0], yy[:, 1], lw=2, label="delay = %.01f"%d)
ax.figure.savefig("lotka.jpeg")The following two outputs are that by this module (left) and ddeint by Zulko (right). The precision of this module's results is higher
There is also an example examples/sine.py which tries to solve a DDE whose solution should be a sine curve to illustrate how numerical errors accumulate over time:
""" Reproduces the sine function using a DDE """frompylabimportlinspace, sin, subplots, pifromddesolverimportsolve_ddedefmodel(Y, t):
returnY(t-3*pi/2) # Modeltt=linspace(0, 50, 10000) # Time start, time end, nb of pts/stepsg=sin# Expression of Y(t) before the integration intervalyy=solve_dde(model, g, tt) # Solvingfig, ax=subplots(1, figsize=(4, 4))
ax.plot(tt, yy)
ax.figure.savefig("sine.jpeg")The following two outputs are that by this module (left) and ddeint by Zulko (right). The precision of this module's results is higher.
Licensed under the MIT license, see LICENSE file for the detailed license text.
ddesolver can be installed by unzipping the source code in one directory, installing setuptools using pip and then running setup.py: ::
(sudo) python3 -m pip install --upgrade pip setuptools
(sudo) python3 setup.py install
You can also install the original ddeint package directly from the Python Package Index with this command: ::
(sudo) pip install ddeint