Sage Interactions - Dynamical Systems
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Cobweb diagrams on [0,1]
by Marshall Hampton.
xxxxxxxxxxdef cobweb(a_function, start, mask = 0, iterations = 20, xmin = 0, xmax = 1): ''' Returns a graphics object of a plot of the function and a cobweb trajectory starting from the value start. INPUT: a_function: a function of one variable start: the starting value of the iteration mask: (optional) the number of initial iterates to ignore iterations: (optional) the number of iterations to draw, following the masked iterations xmin: (optional) the lower end of the plotted interval xmax: (optional) the upper end of the plotted interval EXAMPLES: sage: f = lambda x: 3.9*x*(1-x) sage: show(cobweb(f,.01,iterations=200), xmin = 0, xmax = 1, ymin=0) ''' basic_plot = plot(a_function, xmin = xmin, xmax = xmax) id_plot = plot(lambda x: x, xmin = xmin, xmax = xmax) iter_list = [] current = start for i in range(mask): current = a_function(current) for i in range(iterations): iter_list.append([current,a_function(current)]) current = a_function(current) iter_list.append([current,current]) cobweb = line(iter_list, rgbcolor = (1,0,0)) return basic_plot + id_plot + cobwebvar('x')def cobwebber(f_text = input_box(default = "3.8*x*(1-x)",label = "function", type=str), start_val = slider(0,1,.01,.5,label = 'start value'), its = slider([2^i for i in range(0,12)],default = 16, label="iterations")): def f(x): return eval(f_text) show(cobweb(f, start_val, iterations = its))
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 xvalspretty_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])