- a simple electronic circuit, developed by Leon Chua, showing chaotic dynamics.
The applet shows a simulation of Chua's circuit, plotting the voltage measured across C1 against the voltage measured across C2. This corresponds to the display on an X-Y oscilloscope with probes connected across these capacitors. The initial values of the parameters used in the applet correspond to the component values in the circuit diagram, and show a simple periodic orbit (oscillation). The transition to chaotic dynamics can be found by carefully decreasing R or C1, (e.g. decrease R in steps of 0.01 to 1.2K). The simulation compares well with what is actually seen on an oscilloscope. Chaos seems to develop via a subharmonic cascade.
If you do not have access to an oscilloscope, you can use the voltage across C1 or C2 as the input to a high input impedance audio amplifier (with the component values shown the frequency of the oscillations is in the audio range). It turns out that the ear is very sensitive to the development of a weak subharmonic. The subharmonic becomes the fundamental an octave below the original tone, and the ear hears the note drop an octave even when the intensity of the new fundamental is very weak. The first two or three transitions in the subharmonic cascade route to chaos, and the onset of chaos (noise!) are very audible.
Circuit equations: rather technical, and only useful if you have IBM TechExplorer (sorry, Windows 95 or NT only!) or can read LaTeX files.