In this section we will get a taste of building mathematical models. It is a pretty limited taste because all you can do here is to modify the parameters of the Duffing oscillator. In our DynaLab program you will be able to design your own systems from the beginning. The Duffing Oscillator Research display is similar to the Duffing Oscillator display we have been using, but it has additional features to allow you more control. in the images on this page we depart from our usual practice of hiding the controls to avoid confusing the image with the actual applet. What you see here are raw screen shots, controls included, because the control settings are helpful in interpreting the images.

Across the top of the display is a row of seven input boxes and three buttons each under its own label. The input boxes are where you can change the values of the following parameters:
1. Forcing Function Amplitude (Ampl)
2. Forcing Function Angular Frequency (Ang Freq)
3. Forcing Function Phase Angle (Phase Ang)
4. Energy Loss Coefficient (Drag)
5. Initial Position (Ini x)
6. Initial Velocity (Ini x')
7. Chronons per cycle (Chr/Cyc)

We have met these parameters before with the exception of the number of chronons per cycle. Remember that we coined the term "chronon" to express the notion of the time increment or "delta t" used in integrating the model differential equation. Rather than set the size of a chronon directly, we specify the number of them in a single cycle of the forcing function. The reason for this is buried in the details of the integration algorithm.

The three buttons in the top row of controls allow some manipulation of the high and low limits of the position and velocity variables. The Open button expands the limits by 25% with each click. The Close button reduces the limits by the same amount. The Reset button restores the initial limits. These limits controls can only be used when the model is not running. Clicking the Action button disables them. Also because in the Position/Velocity vs. Time view the limits include the length of the time axis, we cannot manipulate the horizontal limits in that view. The remaining row of controls are those we used in previous displays.

Let's run the research model beginning with the setup for the chaotic attractor we have been using. A screen shot from that run is shown at the right.

Now here is a trick you should always remember to use when exploring a new dynamical system. We set the number of chronons per forcing function cycle at 100 initially so as to not have to wait too long for some results. The fact that a chronon has a finite size in model time when in real time it is apparently very short or perhaps zero, means that the model behavior may be altered by this artificiality. To check this, after you have first run a model, increase the chronons per cycle by a factor of 10, in this case to 1000. If you see the same behavior then you may be reasonable certain that what you are observing comes from the system itself and not from the model. The next run of the model is identical to the previous one except for that change in the number of chronons per cycle.

If you ran the model as set up you probably just saw a message box, which said basically that your model, was out of control. This box comes up whenever your model variables get to more than a million times the total span of the viewing window limits. In this case the viewing limits are set at plus and minus 8 so the first time the program detects a value of x or x' greater in absolute value than 1.6e7 the alarm goes off and the message box comes up. The reported values are those which exceeded the threshold.

You may clear the message box by clicking on Reset, the model initial conditions are restored including setting time back to zero. The 1.6e7 notation by the way represents 1.6 times 10 raised to the 7th power. This is the computer's way of handling scientific notation.

The purpose of this halting the model is that once a model becomes unstable, in only a few chronons it can reach extremely high numbers, beyond the ability of the personal computer to recognize. If the program is running locally, not off the www, this does nasty things to the PC including possibly stepping on some memory in use by the operating system. This would be more aggravation than the average user would be happy with, requiring you to re-start your computer with the loss of any data not written to the disk. By setting the "damage trigger" relatively low we avoid those interruptions to smooth operation. If running this as a Java applet, the browser protects the operating system. In any event, if you are out of bounds by a factor of a million, chances are your model is bogus in some way.

Just to get a feel for how rapidly things can get out of hand, if you did not take notice when the warning message surprised you take a look at the image here and notice the reported values on the message box. In absolute value they are roughly 1e14, which is well over 1e6 times the threshold value. That means that in the last chronon, the model values jumped by at least that factor.

On the bad news box there are a few suggestions as to what sort of things can cause model instability. One of those is that the size of a chronon may be too large. When we changed the angular frequency of the forcing function from 1 to 0.1, we increased the time to complete a cycle by a factor of 10. Since the size of a chronon is set by specifying the number of chronons per cycle, the chronon size was also increased by a factor of 10. The way to keep an eye on this is to watch the dt = statement in the bottom margin. In general we need to keep this under about 0.1 time units. Let's go back and fix the model, setting the angular frequency at 0.1 but increasing the number of chronons per cycle to 1000. You may see the effect by running the modified model.

Now let's go the other way with the frequency of the forcing function. We restore the energy loss coefficient to .05 and put the number of chronons per cycle back to 100. Then make the angular frequency 10. Run this new model.

You begin to see some of the power of using mathematical models. In a few minutes you have learned what it would take weeks and thousands of dollars to learn in a mechanical lab. Let's go back to the model and restore the original parameters of angular frequency = 1, energy loss coefficient of 0.05 and chronons per cycle of 100. Then in the Poincaré Section view we will see the cross-section of the attractor sliced at time equal zero with zero phase angle. The next model is set up to be centered on the cross-section and zoomed so that the image nearly fills the drawing area.

In the next model the only change is to introduce a phase angle of pi/2 = 1.57 radians. Remember that the argument of the cosine in the forcing function is angular frequency times time, plus the phase angle so any non-zero phase angle has the effect of shifting the origin of the time axis. Compare the two Poincaré sections to the two images in the orbit section view above, one at the (p,v) plane and one 90 degrees from it.

Now if you would, go back to the base model except we have put in an energy loss coefficient of 0.085 and chronons per cycle of 1000.

If you have followed my suggestions so far, you have seen some of the techniques by which you can manipulate a mathematical model to make predictions about the underlying physical reality. The mathematics of chaos plays a major role in other dynamical systems also, including growth and decay of ecological systems, competition for scarce resources and the beating of a human heart. We will leave the exploration of these ideas for another course, DynaLab.

For the last time, are there any questions?