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+1=1- mx_n²

• 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 m₁ and m₂ between 0 and 2, m₁ < m₂, Step width of Delta Mu->a value <(m₂ - m₁)/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 x₀ and the factor q for the exponential increase of the deviation as function of n. Choose as the start value St_va= x₀+1E-10 and determine the factor q from the computer experiment.
• Make a "phase diagram" in form of a table or plot for the different regimes of frequencies.
• Search for fractal behavior.

Rigid Rotor

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).

Examples: The program may be started with the input parameters:

· “5,.35,1,.001” and d=0. The RR has several interesting regions: For (Q,w)=(2.2668977,-2.8810204) one finds 32 fix points (FPs), for (Q,w)=( .02698949, 3.025153) one observe 5 FPs, for (Q,w)=(-1.204253, 5.933161) one observe 5 FPs, both set surround the single FP at (Q,w)=( .0592479, 4.807961). There is another center at (.8645298,-1.147262) with one FP. It is surrounded by six FPs with (.4425869,-1.436509), by seven FPs with (-.501359,-2.447427), Sourrounding the single FP at (3.088642,-4.300383), there are a set of five FPs with (-2.792857,-6.032913) at the side and another set of 32 with (2.2668977,-2.8810204).

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