## Thursday, December 13, 2007

### Cobwebs

This is mostly a test of how code looks in a post. The sage functions below are some initial attempts to write basic dynamical systems plotting functions for educational purposes.

def 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)

Note: This is very slow with symbolic functions.
'''
basic_plot = plot(a_function, xmin = xmin, xmax = xmax)
id_plot = plot(lambda x: x, xmin = xmin, xmax = xmax)
iter_list = []
current = start
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)
return basic_plot + id_plot + cobweb

def orbit_diagram(a_function,parameter_interval, domain=[0,1], mask = 50, iterations = 200, param_num = 500.0):
'''
Returns a plot of the iterations of a function as a function of a parameter value.

INPUT:
a_function: a function of one variable
parameter_interval: a two-element list of the lowest and highest parameters to plot.
domain: (optional) a two-element list of the lowest and highest input values to iterate
mask: (optional) the number of initial iterates to ignore
iterations: (optional) the number of iterations to draw, following the masked iterations

EXAMPLES:
sage: f = lambda x,m: m*x*(1-x)
sage: show(orbit_diagram(f,[3.4,4], mask = 100, iterations = 500), xmin=3.4, ymin=0)

NOTES:
This is pretty crude so far.
'''
point_list = []
plen = RDF(parameter_interval[1] - parameter_interval[0])
seed = random()*(domain[1]-domain[0])+domain[0]
for i in srange(parameter_interval[0],parameter_interval[1],plen/param_num):