Procedure
and Tasks

The
experiment consists of two different topics: the **rigid rotor** and the **histogram.
**For both projects special software has been developed.

**Histogram**

- This
program explores the chaotic behavior of the series:

**x _{n+}**

- Load
the computer language HPBasic from the desktop
- Load
the program
**Histogram.hp**from the directory: C:\0aa\SenLab\Experiments\Chaos by typing first**h**, then climbing up the directory with**f2**and finally selecting with**f1**Histogram.hp - The
program starts running. Choose
**CALCULATIONS,**then**PARAMETER**and press return for all suggestions. - Now
you have the options: SINGLE RUN, SWEEP, SERIES.
They are operated according to the following rules.
**SINGLE RUN**,**WITH NUMBERS**->1,**NUMBER OF OPERATIONS**->10000,**NEW Mu**->a value for m between 0 and 2, when question mark appears-> press return. If there are stable values for x_{n}then they are printed out ordered according to their value and according to their appearance. The program continues with**NEW Mu**-. If you want to change to another sub-routine choose the m -value 0.**SWEEP Mu**,**Mu****-values: Start, End**->choose m_{1}and m_{2}between 0 and 2, m_{1}< m_{2}, Step width of Delta Mu->a value <(m_{2}- m_{1})/1000,**NUMBER OF OPERATIONS**->10000. The screen shows the time traces, one after the other, with the corresponding m. Press**Pause**(upper right key) if you want to take notes about a non-chaotic value of m. The subroutine returns to the menu when it has covered the desired range of m.**SERIES**, The subroutine increases m from 0 to 2 in steps of 0.01 and draws in a three dimensional plot the relative occurrence of a value for x_{n}in the x-m-plane.

- Perform
the following tasks:
- Investigate
the full range of m and determine the critical m-values for frequency doubling.

- Search
in the chaotic regime for islands of stability.
- Explore
the behavior for interesting values of m. As some
examples try m=1.631205 which yields 200 different frequencies
or m=1.6243969 which demonstrated the very slow
instability and the role exchange between different frequencies.
- For
a given .75<m<1.25 calculate the instable solution for
**x**and the factor q for the exponential increase of the deviation as function of n. Choose as the start value St_va=_{0}**x**+1E-10 and determine the factor q from the computer experiment._{0} - Make
a "phase diagram" in form of a table or plot for the different
regimes of frequencies.
- Search
for fractal behavior.

This program has a number of commands:

- Start
with parameters, which has the input of
“excitation frequency, amplitude, resonance frequency, damping”. The
resonance frequency is set to 1 which means that the excitation frequency
is given in units of the resonance frequency. The amplitude is the
amplitude of the axis of the rigid rotor (RR) in units of the pendulum
length. The damping is not very important and generally set to 0.001 (it
moves the rotor in the stable range towards the fix point). The next
parameter input is the angle d. For d=0 the axis moves along the x-axis and for d =p/2 it moves
along the y-axis.
- The
run command let the pendulum move. The points on the screen are a plot of w=d Q /dt versus Q. There are
several options to control the program. These options can be activated by
pressing the corresponding key (preferred) or clicking with the mouse.
- 0: Exit
- 1: New start. If one presses (1) once
it clears the window and starts a new run. But the previous points are not
lost. When (1) is pressed again one can choose how many of the old points
shall be conserved. It displays the number of previous points and one can
keep the number, reduce it or set it equal to 1. Enter restarts the run.
- 2: New parameter input (except the
angle d).
- +,-: Changes the scale of Q and w=d Q /dt.
- The
arrows move the plot in the direction indicated by the arrow. Shift arrow
yields a smaller shift.
- 3:
Alternates between original
scale and changed scale.
- 4:
Alternates between continuous
plot and a flash shot after each period of the excitation frequency.
- 5:
Analyses and determines a set
of multi-fix points. You have to locate a set of fix points and then input
5. The program determines the position of the multi-fix points with an
accuracy of 10
^{-6}and prints them out into a file “FixPts.dat” from where they can be copied. (It is in the same directory as the Chaos.hp program. - 6: Not in use with net work printer.
You can use instead the Prt Scr
Sys Rq key and paint.
- 7: Plot can be move from initial
position (mouse click) to final position (mouse click).
- 8: Program idles. Second click
continues program.
- 9:
Numerical input of shifted
origin from the keyboard. Use for the input the desired (Q,w) pair
and click with the mouse at the origin.
- Mouse
click generates a new initial value for (Q,w).

Now
change the amplitude by a very small amount and observe what happens to the FPs. When does their number reduce or they disappear.