THE MODES OF OPERATION:
The period doubling bifurcation software has three modes of operation in which a
certain graphical picture is produced. These pictures are (1) a bifurcation diagram of
the iterative map f, where
f(x) = 1 - ux2 with u as a variable parameter; (2) a
histogram of the trajectory with one specified u value of the initial point x0 = 0 on the
interval [-1,1], or some subinterval of [-1,1]; (3) a visual approximation of what the
"attractor" looks like on the interval.
PRODUCING A BIFURCATION DIAGRAM:
To produce a bifurcation diagram the user must input valid numbers into the text
fields labeled: "u minimum", "u maximum", "y minimum", "y maximum", and "iteration
limit". A bifurcation diagram is then drawn when the button "new diagram" is clicked.
The text fields "u minimum" and "u maximum" receive the minimum and maximum values
intended for the variation of u. In this picture, the u axis is the horizontal axis.
The fields "y minimum" and "y maximum" take the desired minimum and maximum values for
the vertical axis. The vertical axis represents the values that the trajectory of x0 = 0obtains when iterated for each individual u. The trajectory of any initial point (here
we always use x0 = 0) in [-1,1] will always stay in [-1,1], so it is recommended that the
"y minimum" and "ymaximum" values are chosen accordingly (ie any value larger than 1 or
less than -1 for the ymax and ymin respectively is not necessary). The text field
"iteration limit" represents how many iterations will be performed for each u in the
sweep from umin to umax (about 350 individual u values). The inputs for the fields that
are considered valid are as follows:
any minimum value must be less than the maximum value for a specific parameter
iteration limit must be greater than 100
example parameter values:
u minimum = 0
u maximum = 2
y minimum = -1
y maximum = 1
iteration limit = 350
PRODUCING A HISTOGRAM:
To produce a histogram for the trajectory of x0 = 0 with one specific u value the
text fields labeled "u =", "histogram points", and "height" must contain valid entries.
Clicking the button labeled "histogram" will produce a histogram with the specified
parameters. The text field "u =" takes the specific u value u would like to look at the
trajectory of x0 = 0 for. The text field "histogram points" represents the number of
points in the trajectory that will be counted to produce the histogram. Simply stated,
"histogram points" is the number of iterations to be performed (a value larger than
100,000 is not recommended, moreover something between 3000 and 20,000 is good). The
text field "height" represents the height of the histogram, or how tall the window is.
Valid inputs for the histogram fields are as follows:
u can be any real number (poorly chosen u values may not produce an interesting picture
though)
histogram points must be greater than 500
height needs to be a positive integer
notes on the histogram:
The horizontal axis of the histogram shows the interval on which the trajectory
of x0 = 0 for the given u stays after 500 iterates. This interval may not necessarily be
the entire interval [-1,1]. For this reason, the smallest interval for which the
trajectory stays in after a certain number of iterates is used to determine the minimum
and maximum values on the horizontal axis. The histogram is a good tool for showing how
much "time" a certain trajectory spends in a certain piece of the interval relative to
others (i.e. for how many iterations was the trajectory near a certain piece of the
interval).
example values:
u = anything in [0,2]
histogram points = 10,000
histo height = 10,000
(note if the spikes on the histogram look very small, simply lower the histo height
value)
PRODUCING A "VIZUALIZATION" OF THE ATTRACTOR:
The "attractor" portion of this software uses the same fields as the histogram
section. So the text fields which need appropriate entries for the "attractor" feature
are the "u=" and "histogram points" text fields. The "u=" field specifies for which u
value the user wants to look at the attractor and the "histogram points" will serve as
the iteration limit for which the attractor will be constructed. Valid inputs are the
same as for the histogram (see the previous section).
notes on the "attractor":
This feature simple shows a picture of an interval with (very small) red hash
marks to indicate if the trajectory visits that piece of the attractor. The endpoints on
the interval display are chosen as the minimum and maximum values the specified
trajectory visits after 100 iterations. The initial points in the trajectory are not
what we are concerned with when looking for the attractor. So the program allows for 100
iterations to pass before keeping track of the long-term behavior of the specific
trajectory.
