Science and Computers I -- PHY307/607
Lab 5. - Modifying the Bifurcate Applet and Fractal
Dimensions
In this lab you will learn how to download, modify and recompile a Java
applet built from several source Java files - the Bifurcate applet
you used before. You will use this to explore the Bifurcation
diagram for another 1D map - the sine mapping. The point about this is
that many features of chaos and the approach to chaos are independent
of the precise nonlinear function used in the dynamics - thus a sine curve
is similar to the logistic parabola. This feature is called
universality.
Next you will use another applet - the LogFractalApplet to
try to estimate the fractal dimension of the strange attractor
governing the chaotic regime of the logistic map.
Modifying the Bifurcate Applet to look at the sine mapping
- Make a directory inside your public_html/PHY307 folder called
Bifurcate
(inside PHY307 use the unix command mkdir Bifurcate).
Change its permissions using the command chmod 755 Bifurcate.
Now download the Java code for the Bifurcate Applet from the
Labs section of the PHY307 course pages to this new directory. You should
find five .java source files. When you select Save As
from within Netscape select the Bifurcate subdirectory of
public_html/PHY307. Once you have done this change to this
directory by typing (from within PHY307) cd Bifurcate .
- We will modify the Map class to compute the bifurcation
diagram not for the logistic map but the sine map. Use
pico to edit the file Map.java. Inside the
iterate method within the class
definition find the line which codes the
iteration x=a*x*(1-x) and replace it with code to compute instead
the map x=a*Math.sin(3.141592654*x).
- Now recompile all the Java source files with the unix
command javac *.java.
- Change the file permissions on the .class files with the command
chmod 644 *.class.
- Next within this new directory create a file (using pico) called
BifurcateApplet.html. This file should contain the line
< applet code=BifurcateApplet.class height=400 width=600 > < /applet >
Remember that there should be no spaces between the < and > signs
and the following word tag.
- Type chmod 644 BifurcateApplet.html
- Point Netscape at the address web.syr.edu/~yourSUnixuserid/Bifurcate/BifurcateApplet.html. You should see the Bifurcation Applet now running off
your own Web page.
- Change a_2 to 1.0 and hit Go. What range of the parameter
a now corresponds to a fixed point value for x ?
- Where (approximately) does the transition to 2-cycle behavior
occur ?
- For what value of a does the system start to exhibit 4-cycles
- What about 8-cycles ?
- Compute an approximate value for the Feigenbaum constant for this
mapping. Comment ...
The fractal dimension of the strange attractor in the logistic map
As we will further discuss in class we can ascribe a fractal dimension
to an object such as a strange attractor by counting the number of
boxes of a certain size needed to cover that object. The
definition of dimension will coincide with the usual notion of
dimension for regular (non-fractal) objects but gives us a useful
additional quantity for classifying fractal objects.
For the 1D maps
we have been studying so far the idea is to take divide the interval
between 0 and 1 into a large number of (non-overlapping) intervals and
count the number of these intervals that contain at least one
point of the attractor (i.e contain a point generated by iteration of
the mapping). We then plot the log of the number of occupied boxes as
a function of the log of the size of the box. If the object is
a fractal this should give a straight line whose gradient is the
fractal dimension. The applet allows you to change the parameter a
and observe how that effects the fractal dimension - the latter is
computed by the applet by fitting a straight line through the
points generated by the counting procedure. Notice the x-axis is
essentially the log of the interval (box) size and the y-axis contains
the log of the number of occupied boxes.
- Go to the Labs section of the PHY307 page and link to the
LogFractal Applet. For a=3.2 sketch what you see when you hit
the reset button.
- Hit Fit. What is the fractal dimension you read off the text
box labelled D ? Try other values of a for a<3.5.
What do you conclude about the fractal dimension
in a non-chaotic regime ?
- Set a=3.56995. This is good approximation for the critical
value of a at which the system becomes chaotic. What is the
fractal dimension there ?
- Set a=3.8. What is the fractal dimension here ? You should see
that the dimension of this attractor depends on a and increases
towards 1 as a increases.
- What happens to the fractal dimension for a=3.84 ? Can you
interpret this in terms of the appearance of a 3-cycle ? (remember where
the 3-cycle appeared in lab 3.
To get credit for this lab you need to
- Answer the questions posed in the lab -- hand in your
answers on paper.
The deadline to get these things to me is next Thursday.
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This page maintained by Simon Catterall, last updated 4 October, 2001.