From the theory page, we saw a bifurcation diagram leading to chaotic behavior in an LRC circuit.  In our circuit, the frequency used was 80 kHz.  This might seem odd because since the resonance of the circuit is 25 kHz.  The reason we used 80 kHz instead is because at 25 kHz the different states leading to chaos were extremely unstable and close together, making analysis and the ability to take pictures nearly impossible.

Here is the same bifurcation chart with the amplitude values (output voltages) that were observed at each state:

To see what the states look like exactly, in an Vin vs. Vout method (a nice way to see the phenomenon), click here.

Each of these moments can be put through a fast Fourier transform (FFT) which will show the patterns (or lack thereof) in the data.  Below are the Fourier transforms of the data at single period, period doubling, period times eight, and chaos.  Unfortunately, the period quadrupling was very unstable and a good scope capture was impossible.

 Fast Fourier Transforms of Scope Screen Captures Single Period The frequencies involved are restricted to discrete, evenly spaced intervals, showing the periodicity of a single period. Period Doubling This is perhaps the most obvious.  The period doubles here, again restricted to very specific frequency values. Period Times Eight The best place to see this one is around 100kHz, where the preceding three frequencies build up to the final highest one, demonstrating period times eight. Chaos There are many frequencies involved, and they have no pattern to them whatsoever-- they are definitely not evenly spaced, showing the chaotic regime.

Loose Ends:

In less chaotic terms, the capacitance of the diode is simple to find using the resonance equation:  .  Since we know L (45 mH) and the resonance frequency (25 kHz), the capacitance of the 1N4001 diode is 35.5 nF.