Sage Interactions - Dynamical Systems

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Cobweb diagrams on [0,1]

by Marshall Hampton.

Cythonized Logistic Orbit Map

By Marshall Hampton

%cython
cpdef double logorb(double k,long N,double x0):
    cdef double x = x0
    cdef long i 
    for i from 1 <= i <= N:
        x = k*x*(1-x)
    return x

cpdef logtraj(double k,long N, double x0):
    cdef double x = x0
    xvals = []
    cdef long i 
    for i from 1 <= i <= N:
        x = k*x*(1-x)
        xvals.append(x)
    return xvals

pretty_print(html('<h2>Orbit diagram of the logistic map</h2>'))
@interact
def logistic_bifs(k_min = slider(0.0,4.0,.001,3.5), k_max = slider(0.0,4.0,.001,4.0)):
    tkmax = max(k_min, k_max)
    tkmin = min(k_min, k_max)
    dk = (tkmax - tkmin)/1000.0
    xpts = []
    x = .5
    for k in srange(tkmin,tkmax,dk):
        x = logorb(k,100,x)
        ks = logtraj(k,12,x)
        if max(ks)-min(ks) < .001:
            xpts.append([k,x])
        else:
            x = logorb(k,1000,x)
            ks = logtraj(k,100,x)
            xpts = xpts + [[k,q] for q in ks]
    show(points(xpts, pointsize = 1), figsize = [6,6])