Making of-the-logistic-map-bifurcation-diagram

Making of:
The logistic map
bifurcation diagram
BradMartsberger

What is this talk about?
Makingaprettypicture
Squeezingperformance outof python
PyPy, Cython, C extensions
With atoyphysics model(chaos!)

Cool, but...
Where did that come from?
Can you show me some python already?

The logistic map
Map avalue, x, from [0, 1] to [0, 1] accordingto:
r *x *(1 -x)
Let's play...
x=0.25
r=2.5
forjinrange(limit):
print"x:",x
x=r*x*(1-x)

What do we need to make that
sweet graph?
Python
Matplotlib
Patience ... lots of patience

Simplified version
bif_diag=[[0.0]*height]*width
forkinrange(width):
#histogramofhowfrequentlyeachpixelisvisited.
h=logmap_histogram(r,x_len,x1,x2,height)
#Normalizethehistogramtogofrom0to1.
h=divide_list(h,1.0*max(h))
#stickitinupsidedowntotheimage
bif_diag[k-1]=h[::-1]
plot_bif_diag(bif_diag,plt,cm,(x1,x2),(r_lower,r_upper))

Inner loop
deflogmap_histogram(r,x_len,x1,x2,height):
result=[0]*height
dx=1.0*(x2-x1)/(height-1)
transient=500
x=0.25
forkinxrange(transient):
x=x*r*(1-x)
forkinxrange(x_len):
x=x*r*(1-x)
incr=int((x-x1)/dx)
ifincr>=0andincr<M:
result[incr]+=1
returnresult

Matplotlib bit
defplot_bif_diag(bif_diag,plt,cm,xlim,rlim):
width,height,=len(bif_diag),len(bif_diag[1])
to_plot=[[1-pixforpixincol]forcolinbif_diag]
to_plot=map(list,zip(*to_plot))
plt.imshow(to_plot,cm.gray,clim=(0.7,1))
font={'family':'serif','size':24}
plt.xlabel('r',fontdict=font)
plt.ylabel('x*',fontdict=font)
plt.xticks(linspace(1,N,9),['%.2g'%rforrinlinspace(rlim[0],rlim[1],9)])
plt.yticks(linspace(1,M-1,5),['%.2g'%xforxinlinspace(xlim[1],xlim[0],5)]
ax=plt.gca()
ax.set_position([80./(width+100.),80./(height+100.),width/(width+100.),
plt.gcf().canvas.manager.window.geometry("{0}x{1}+30+30".format(width+100,height+
plt.show()

Let's run this
thing already...

That was a little disappointing
Nothorrible, buttoo slow to do some coolstuff.

Options for doing better
0. Quitusingpython
1. PyPy
2. C extension
3. Cython
4. There are others (numpywith numexpr)

PyPy
Straightfrom pypy.org:
Fast, compliantalternative implementation of the Python
language (2.7.6 and 3.2.3) with severaladvantages:
Speed, due to Just-In-Time compiler
Takes less memory
Supports anumber of third partylibraries:
ctypes, django, sqlalchemy, twisted, etc.
Does notrequire anychange to your code!
Unfortunately, matplotlib is notsupported.
Numpyis partiallysupported.

C extensions
Write functions in C thatare callable from python
Pass python objects as arguments thatmustbe converted to C
values
Returns apython object, e.g., alist
Compiled and fast
Do this atleastonce

Let's have a look
staticPyObject*logmap_attractor_histogram(PyObject*self,PyObject*args){
doubler1,r2,x,x1,x2;
intxxlen,rrlen,M;
inttransient=500;//Shorttransientforeachr.
if(!PyArg_ParseTuple(args,"ddiiddi",&r1,&r2,&rrlen,&xxlen,&x1,&x2,&M)){
returnNULL;
}
//Storeourhistograminanintarray,copyittoapythonlistandreturn
//thelistasaPyObject*
intarray[M];
PyObject*list=PyList_New(M);
//Initializearraytoallzeros
intj,k;
for(j=0;j<M;++j){
array[j]=0;
}
doubler,dr,dx;
intincr;
//Settheamountwewillsteptogetfromr1tor2inrrlen-1steps
dr=0;
if(rrlen>1){dr=(r2-r1)/(rrlen-1);}
//Histogrambinsize
dx=(x2-x1)/(M-1);
//loopoverr(perhapschangetofor(r=r1;r<=r2;r+=dr))

Compiling to an importable module
Create asetup.pyfile, e.g:
fromdistutils.coreimportsetup,Extension
setup(name='logmap',version='1.0',
ext_modules=[Extension('logmap',['logmapmodule.c'])])
Then installit
>>>pythonsetup.pyinstall
And itcan be imported and called
importlogmap
h= logmap.attractor_histogram(...)

Cython
Directlyfrom cython.org:
Static compiler for python
Makes writingC extensions for python as easyas writing
python
Generates C code from your python code
Has the potentialto deliver the speed gains thatwe getfrom aC
extension with less hassle

How to make it work
Create setup.pyfile:
Build like this:
Then you can importfrom
logistic_map_bifurcation_diagram_cython
fromdistutils.coreimportsetup
fromCython.Buildimportcythonize
setup(
name='logmap_cython',
ext_modules=cythonize("logistic_map_bifurcation_diagram_cython.pyx"),
)
%>pythonsetup.pybuild_ext--in_place

Making cython faster
Cython's code generation is helped byprovidingithints in the
form of type declarations
deflogmap_histogram(floatr1,floatr2,intrrlen,intxxlen,floatx1,floatx2,intM)
result=[0]*M
cdeffloatdx
dx=1.0*(x2-x1)/(M-1)
transient=500
cdeffloatx
cdeffloatr
cdeffloatk
forrinlinspace(r1,r2,rrlen):
x=0.25
forkinrange(transient):
x=x*r*(1-x)
forkinrange(xxlen):
x=x*r*(1-x)
incr=int((x-x1)/dx)
ifincr>=0andincr<M:
result[incr]+=1
returnresult

Making of-the-logistic-map-bifurcation-diagram

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