Science and Computers I -- PHY307/607

Lab 3. - Java Buttons and Period Doubling

In the first part of this lab you will learn how to add a clickable button to your HelloWorldApplet. You will then use another Java applet - the BifurcateApplet to explore the approach to chaos in the logistic map.

Adding buttons to the HelloWorldApplet

Point Netscape at the PHY307 homepage and look under the Labs section. You should see a link to the code Hwa3.java. Download this to your SUnix account by selecting Save As from the file menu and choosing to save it to directory/folder public_html/PHY307 on the I: drive of your PC.
Open a Telnet session to SUnix as before and go to the directory public_html/PHY307 (by typing cd public_html/PHY307).
Start up an editing session of this new file by typing pico Hwa3.java. You will see that the code is pretty much the same as that in Hwa2.java. We will start to edit it now to add a button which controls what is drawn to the screen.
After the line private Font newfont;, add the lines
```
private Button mybutton;
private boolean onbutton=false;
```
By doing this you have defined two new objects in your code -- the object mybutton which is of type Button and a logical variable onbutton which is either true or false depending on whether mybutton has been clicked. It is initially set to false.
Inside the init() method (that is between the curly braces defining the content of the method) and directly after the line newfont = new Font ... add the lines
```
mybutton= new Button("Press Me");
add(mybutton);
```
This creates the Button mybutton (with the special new command) and labels it Press Me. It then adds it to the applet page with the add command.
Finally, notice that a new method action() has been added to the applet code. This method watches for clicks of mybutton and responds by flipping the logical variable onbutton alternately from true to false. The words Hello World are only displayed when onbutton is true.
The code to do this is listed below. Add these lines between the beginning and ending brace of the action() method.
```
if(evt.target instanceof Button){
onbutton=!onbutton;
repaint();
return true;
}
return false;
```
Recompile the code by executing the command javac Hwa3.java at the SUnix prompt.
Change the file permissions as usual by typing chmod 644 Hwa3.class.
Now use pico to create a page - called say Hwa3.html which will load this new Java applet Hwa3.class. Remember to set its permissions correctly.
Fire up Netscape and point it at this new page.
IMPORTANT- Sometimes Netscape caches copies of Web pages and Java applets on disk so that hitting reload by itself does not always get the most recent version of the applet off the network. Try holding down the shift key at the same time as hitting reload - this (sometimes!) forces a reloading of Java code for sure. If this doesn't work try explicitly flushing the cache - look under the Preferences/Advanced menu in Netscape. If all else fails close and reopen Netscape or fire up Explorer instead.

What do you see ?
What happens when you click on the button ?

Period Doubling as a route to chaos

In Netscape go to the PHY307 homepage. Look under the LABS section and link to the Java applet Bifurcation Applet. You should see a graph appear within your browser. Hit the Go button.

The plot shows the possible values of the variable x seen under iteration of the logistic map we have discussed in class (and which you studied in lab. 2). This map takes the form
```
x_n+1=ax_n(1-x_n)
```
The possible values of x are plotted as a function of the value of the parameter a. Over what region of a is a single value of x seen ? This is sometimes called a 1-cycle or fixed point.
How many values of x are seen just to the right of this region ? The values of x in this nearby region form a 2-cycle. The possible values of x alternate between two values i.e the period of oscillation is 2. Fixed points have periods of magnitude 1. Thus the transition between the two is called a period doubling transition.
Now we will look closely at the region close to a=3.0. To do this type the value 2.9 into the box labelled by a_1= and set a_2=3.1. Remember to hit return after typing each of the numbers. Now hit Go again. Estimate the value of a at which the period doubling occurs. Call this value p_1.
Now we will look closely at the region close to a=3.4. Set a_1=3.4 and a_2=3.5. Remember to hit return after typing each of the numbers. Now hit Go again. What happens to the curves close to x=3.44 ? We say that the 2-cycle has split into a 4-cycle via another period doubling transition. This splitting is sometimes called a bifurcation (hence the name of the applet!). Estimate the value of a where this happens - call it p_2.
Finally look at the region near a=3.5. Set a_1=3.5 and a_2=3.56. Locate the approximate value of a at which another period doubling transition occurs. Call this value p_3.
Compute the quantity
```
(p_2-p_1)/(p_3-p_2)
```
This will turn out to be an important universal number called Feigenbaum's delta. The significance of this number will be discussed in class.
Finally, set a_1=3.6 and a_2=3.9. Hit Go. The screen should be almost full of points as you are in the middle of a chaotic region where seemingly all values of x are attained during iteration of the map. But look closely in the region of a=3.8-3.85. What do you notice ? How many values of x are seen there ? This is an example of a periodic window. Suddenly, in the midst of chaos order reemerges...
To get credit for this lab you need to
1. Email me the url to your Web page so that we can take a look at your working Hwa3 applet.
2. 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 13 September, 2000.