pydelay-master_DDE__Lang-Kobayashiequations资源-CSDN文库

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"pydelay-master_DDE_" 指的是一款基于Python的库，专门用于解决延迟微分方程（Delay Differential Equations, DDEs）的问题。延迟微分方程在许多科学和工程领域都有应用，例如生物系统、控制系统理论、化学反应动力学等。PyDelay是一个强大的工具，它允许用户方便地定义和求解这类复杂方程。中提到的“差分方程计算”是指使用数值方法来解决数学中的差分方程，尤其是延迟微分方程。Lang-Kobayashi方程组是量子光学领域中的一组重要方程，描述了非线性光学系统，如半导体激光器或光学参量振荡器的行为。这个方程组包含了复杂的相互作用和时间延迟效应，通常需要数值模拟来得到实际的解决方案。PyDelay-master项目提供了解决这类问题的框架，使得研究人员和工程师可以更有效地分析和理解这些系统的动态行为。 "DDE" 明确了这个项目的重点在于处理延迟微分方程。在DDEs中，一个变量的值不仅依赖于当前时间点的状态，还取决于过去的值，这增加了求解的复杂性。PyDelay库通过提供一系列内置的数值方法，如BDF（Backward Differentiation Formulae）和LSODA（Livermore Solver for Ordinary and Delay Differential Equations with Automatic Selection），来应对这种挑战。【压缩包子文件的文件名称列表】: "pydelay-master" 暗示了这是一个GitHub仓库的下载，其中包含了整个项目的所有源代码和其他资源。在这个目录下，可能包含如下内容： 1. `setup.py`：Python项目的安装脚本，用于构建和安装PyDelay库。 2. `README.md`：项目介绍和使用指南，可能包含示例和安装步骤。 3. `src/` 目录：存放源代码，包括核心算法和函数实现。 4. `tests/` 目录：测试用例，用于验证库的功能和性能。 5. `examples/` 目录：可能包含一些演示如何使用PyDelay解决特定DDE问题的实例。 6. `docs/` 目录：可能包含文档，如API参考和用户手册。通过这个库，用户可以定义自定义的DDE问题，设置初始条件和边界条件，选择合适的数值方法，然后运行模拟来获取结果。PyDelay可能还提供了可视化功能，帮助用户直观地理解和分析结果。对于那些需要研究含有时间延迟效应复杂系统的科学家和工程师来说，PyDelay是一个强大的工具，能够简化他们的工作流程，提高研究效率。

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pydelay-master.zip （24个子文件）

pydelay-master

pydelay

helper.py 4KB

_dde23.py 43KB

__init__.py 296B

doc

pydelay.pdf 217KB

conf.py 7KB

sphinxext

ipython_console_highlighting.py 4KB

inheritance_diagram.py 13KB

apigen.py 15KB

numpydoc.py 4KB

docscrape.py 14KB

docscrape_sphinx.py 4KB

pyplots

lk.py 1KB

mackey-glass.py 1009B

index.rst 10KB

Makefile 3KB

README 149B

examples

lk.py 1KB

mackey-glass.py 505B

ccode_example.py 821B

.man.py.swp 12KB

fhn.py 1KB

LICENSE 1KB

setup.py 499B

.gitignore 27B

pydelay – a python tool for solving delay differential equations

v0.1.1

V. Flunkert E. Schöll

November 09, 2009

Contact: ﬂunkert@itp.tu-berlin.de

1 Introduction

pydelay is a program which translates a system of delay differential equations (DDEs) into simulation C-code and

compiles and runs the code (using scipy weave). This way it is easy to quickly implement a system of DDEs but

you still have the speed of C. The Homepage can be found here:

http://pydelay.sourceforge.net/

It is largely inspired by PyDSTool.

The algorithm used is based on the Bogacki-Shampine method

which is also implemented in Matlab’s dde23

We also want to mention PyDDE – a different python program for solving DDEs.

License

pydelay is licensed under the MIT License.

1.1 Installation and requirements

Unix:

You need python, numpy and scipy and the gcc-compiler. To plot the solutions and run the examples you also

need matplotlib.

To install pydelay grab the latest tar.gz from the website and install the package in the usual way:

cd pydelay-$version

python setup.py install

When the package is installed, you can get some info about the functions and the usage with:

pydoc pydelay

Windows:

The solver has not been tested on a windows machine. It could perhaps work under cygwin.

Bogacki, P. and Shampine, L. F., A 3(2) pair of Runge - Kutta formulas, Applied Mathematics Letters 2, 4, 321 ISSN 0893-9659, (1989).

Shampine, L. F. and Thompson, S., Solving DDEs in Matlab, Appl. Num. Math. 37, 4, 441 ( 2001)

arXiv:0911.1633v1 [nlin.CD] 9 Nov 2009

1.2 An example

The following example shows the basic usage. It solves the Mackey-Glass equations

for initial conditions which

lead to a periodic orbit (see

for this example).

# import pydelay and numpy and pylab

import numpy as np

import pylab as pl

from pydelay import dde23

# define the equations

eqns = {

’x’ : ’0.25

x(t-tau) / (1.0 + pow(x(t-tau),p)) -0.1

x’

}

#define the parameters

params = {

’tau’: 15,

’p’ : 10

}

# Initialise the solver

dde = dde23(eqns=eqns, params=params)

# set the simulation parameters

# (solve from t=0 to t=1000 and limit the maximum step size to 1.0)

dde.set_sim_params(tfinal=1000, dtmax=1.0)

# set the history of to the constant function 0.5 (using a python lambda function)

histfunc = {

’x’: lambda t: 0.5

}

dde.hist_from_funcs(histfunc, 51)

# run the simulator

dde.run()

# Make a plot of x(t) vs x(t-tau):

# Sample the solution twice with a stepsize of dt=0.1:

# once in the interval [515, 1000]

sol1 = dde.sample(515, 1000, 0.1)

x1 = sol1[’x’]

# and once between [500, 1000-15]

sol2 = dde.sample(500, 1000-15, 0.1)

x2 = sol2[’x’]

pl.plot(x1, x2)

pl.xlabel(’$x(t)$’)

pl.ylabel(’$x(t - 15)$’)

pl.show()

Mackey, M. C. and Glass, L. (1977). Pathological physiological conditions resulting from instabilities in physiological control system.

Science, 197(4300):287-289.

http://www.scholarpedia.org/article/Mackey-Glass_equation

0.2 0.4 0.6 0.8 1.0 1.2

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1.6

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)

0.2

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0.6

0.8

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−

15)

2 Usage

2.1 Deﬁning the equations, delays and parameters

Equations are deﬁned using a python dictionary. The keys are the variable names and the entry is the right

hand side of the differential equation. The string deﬁning the equation has to be a valid C expression, i.e., use

pow(a,b) instead of a

b etc.

Delays are written as (t-delay), where delay can be some expression involving parameters and numbers but

not (yet) involving the time t or the dynamic variables:

eqns = {

’y1’: ’- y1

y2(t-tau) + y2(t-1.0)’,

’y2’: ’a

y2(t-2

tau) - y2’,

’y3’: ’y2 - y2(t-(tau+1))’

}

Complex variables can be deﬁned by adding ’:c’ or ’:C’ in the eqn-dictionary. The imaginary unit can be used

through ’ii’ in the equations:

eqns = {

’z:c’: ’(la + ii

w0 + g

pow(abs(z),2) )

z + b

(z(t-tau) - z(t))’,

}

Parameters are deﬁned in a separate dictionary where the keys are the parameter names, i.e.,:

params = {

’a’ : 0.2,

’tau’: 1.0

}

2.2 Setting the history

The history of the variables is stored in the dictionary dde23.hist. The keys are the variable names and there

is an additional key ’t’ for the time array of the history.

There is a second dictionary dde23.Vhist where the time derivatives of the history is stored (this is needed for

the solver). When the solver is initialized, i.e.,:

dde = dde23(eqns, params)

the history of all variables (deﬁned in eqns) is initialized to an array of length nn=101 ﬁlled with zeros. The

time array is evenly spaced in the interval [-maxdelay, 0].

It is possible to manipulate these arrays directly, however this is not recommended since one easily ends up with

an ill-deﬁned history resulting for example in segfaults or false results.

Instead use the following methods to set the history.

hist_from_funcs(dic, nn=101)

Initialise the histories with the functions stored in the dictionary dic. The keys are the variable names. The

function will be called as f(t) for t in [-maxdelay, 0] on nn samples in the interval.

This function provides the simplest way to set the history. It is often convenient to use python lambda

functions for f. This way you can deﬁne the history function in place.

If any variable names are missing in the dictionaries, the history of these variables is set to zero and a

warning is printed. If the dictionary contains keys not matching any variables these entries are ignored and

a warning is printed.

Example: Initialise the history of the variables x and y with cos and sin functions using a ﬁner sampling

resolution:

from math import sin, cos

histdic = {

’x’: lambda t: cos(0.2

t),

’y’: lambda t: sin(0.2

}

dde.hist_from_funcs(histdic, 500)

hist_from_arrays(dic, useend=True)

Initialise the history using a dictionary of arrays with variable names as keys. Additionally a time array can

be given corresponding to the key t. All arrays in dic have to have the same lengths.

If an array for t is given the history is interpreted as points (t,var). Otherwise the arrays will be evenly

spaced out over the interval [-maxdelay, 0].

If useend is True the time array is shifted such that the end time is zero. This is useful if you want to use the

result of a previous simulation as the history.

If any variable names are missing in the dictionaries, the history of these variables is set to zero and a

warning is printed. If the dictionary contains keys not matching any variables (or ’t’) these entries are

ignored and a warning is printed.

Example::

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