Portable Real-Time Applications
With Tcl/Tk, it is so simple to implement stage 1 that we can afford to look at two different implementations. When executing the script in Listing 2 with the wish -f duffing.tcl command, the two independent parameters k and B are visualized as the axes of a two-dimensional coordinate system. Drag the circle to any place on the map, and the new coordinates will be printed to standard output. But maybe you prefer the simpler implementation shown in Listing 1, which displays both parameters as scales. If you run a computer without a GUI, you can still work with the software presented here. In this case, forget about stage 1; stage 2 will read its input from the command line. Enter lines like k 0.05 or B 7.5 at run time, and stage 2 will not notice the difference.
Stages 2 and 3 had to be integrated into one program for reasons of efficiency. It was tempting to implement each stage as a single thread of execution. Threads of execution behave mostly like processes that share a single data space: one waiting for input to modify parameters, the other one calculating the wave form to be emitted. How does one implement threads of execution in a portable way? The POSIX thread library (see Resources) is now available for most operating systems, including the ones mentioned earlier. For example, at STN Atlas Elektronik we use threads for sound generation with multiple sound cards in a multiprocessor setting (two CPUs and Linux 2.2 SMP). As explained in David Butenhof's excellent book, threads make it very hard to debug software; therefore, we will refrain from using them here.
Fortunately, the problem of dealing with unsynchronized events can be solved with the often underestimated select system call (Listing 4, function main). The main loop of Listing 4 has a short loop to check whether there is data coming in from standard input. Then, it calculates a block of data by Finite Differencing and finally emits it. While calculating, some data points are printed to standard output. Only those which occur at integral multiples of the cycle period of the driving force, having the same phase angle, are printed. This technique of selecting data points to display is at the heart of the Poincaré section, a kind of stroboscope which reveals hidden order within chaotic data (Figure 5). Notice that this data is the input to stage 4.
Figure 3. The default parameters show sensitive dependence on initial conditions (see page 4, Thompson/Stewart).
This was the easy part; the hard part is handing over the data to the sound system in a portable way. In this respect, Linux is the platform handled most easily. Writing data to the special file /dev/dsp is enough. With IRIX, we also need just one function call; not the usual write, but a special sound function. In both cases, synchronization is implemented by the blocking behaviour of these functions. This is in contrast to Win32, which bothers the programmer with buffer handling, and synchronization must be done with callback functions. Using the new DirectSound API was not an option because Microsoft has failed to implement the DirectX API in Windows NT.
Stage 4 is, again, rather simple to implement (Listing 5, Figure 5). Each data point read is printed as a dot in the phase space diagram. When producing Poincaré section data as in stage 3, linear oscillators produce circles or spirals, degenerating into fixed points, which is rather boring. A chaotic oscillator is needed to plot the strange attractor shown in Figure 5.
Figure 4. High resolution Poincaré section of one chaotic attractor