Before discussing what this form does, let me define some terms for you, the uneducated reader (or the educated grader). Iteration is the repeated application of a transformation. For example, imagine that you have a generic function y = f(x), and choose an initial point x₀ in the domain of f. Now calculate f(x₀) and set x₁ equal to the result. From then on, each repeated iteration gives a new term x_n+1 = f(x_n). This is the wonderful process known as iteration.

Another important term is bifurcation. I won't describe bifurcation here, but if you want to see own definition then you can go to my bifurcation program page, which has all the info you need. You can use this script to find examples of bifurcation. "How?" you ask. That's an excellent question! Well, you use it as described below:

So what exactly does the above form do? It takes any of four given functions, a starting x-value, a starting k-value, and a number of iterations (n [1, 1000]), and it does that many iterations. Well, that may not be too specific, so here's a slightly better explanation: each of the possible functions has two parameters, k and x₀. You choose those, obviously, and then the iterations are calculated by x_n+1 = f(x_n), where f is the chosen function. It then prints out each of the iterations on my PHP-driven page. I won't give a link to that because it returns an error if there's no valid data given, so if you want to see that in action, fill out the above form and click "Submit."

Another thing worth trying to find is a cycle, where a series of values is repeated every few iterations. The default k-value for the logistic equation leads to a 3-cycle (a cycle of length 3), where the iterations converge approximately to the x-values 0.15, 0.49, and 0.96. Experiment and see what else you can find!

<FORM name="data" ACTION="parabola.php3" METHOD=POST>

<!-- SELECT: CHOOSE FUNCTION -->
<P>The function:<BR>
<SELECT name="function">
<OPTION>y = k*x*(1-x)
<OPTION>y = sin(k/x)
<OPTION>y = k*e^(cos x)
<OPTION>y = k*sqrt(x)
</SELECT></P>

<P>

<TABLE border=1 bgcolor="#CCCCCC"><TR><TD valign=center>

<TABLE border=0 valign=center>

<!-- TEXTBOX: GET K-VALUE -->
<TR>
  <TD align=right><FONT color="#000000">
  <B><I>k</I></B> value:
  </FONT></TD>

  <TD align=left>
  <INPUT TYPE=text NAME="kval" VALUE=3.841499007543>
  </TD>
</TR>

<!-- TEXTBOX: GET X-VALUE -->
<TR>
  <TD align=right><FONT color="#000000">
  <B><I>x<sub>0</sub></I></B> value:
  </FONT></TD>

  <TD align=left><INPUT TYPE=text NAME="xval" VALUE=0.1>
  </TD>
</TR>

<!-- TEXTBOX: GET NUMBER OF ITERATIONS -->
<TR>
  <TD align=right><FONT color="#000000">
  Iterations:
  </FONT></TD>

  <TD align=left>
  <INPUT TYPE=text NAME="nIter" value="10">
  </TD>
</TR>

<!-- SUBMIT FORM DATA! -->
<TR>
  <TD colspan=2 align=center>
  <INPUT type=submit value="Submit">
  </TD>
</TR>

</TABLE>

</TD></TR></TABLE>

</P>

</FORM>