Synchronized Chaos: Electronic Circuit Implementation of the Lorenz Equations

Linnea Engstrom '05
Advisor: David Jackson

This project is an attempt to replicate work done by Kevin Cuomo as published in Phys. Rev. Lett. 71, 65 (1993) and outlined in Section 9.6 of the text by Steven Strogatz, "Nonlinear Dynamics and Chaos," (Addison Wesley, Reading, MA, 1994).

We begin by constructing an electronic circuit using operational amplifiers configured as integrators to replicate the Lorenz Equations.  The pictures below show oscilloscope screen-shots when viewing two of the three output signals in x-y mode.  This demonstrates that we have a chaotic electronic circuit that reproduces the Lorenz equations.

Then, a second, nearly identical circuit is constructed and one of the outputs of the first circuit is used as an input to the second circuit.  The result is that the two circuits become synchronized and produce (nearly) identical chaotic signals.  The picture to the right shows an oscilloscope screen-shot when viewing one of the outputs from each of the two circuits in x-y mode.  The straight line demonstrates that the two chaotic signals are very well synchronized.

Lastly, these synchronized circuits can be used to send "secret" messages.  The first circuit is used to mask an audio signal (music, for example) with high amplitude chaotic noise.  This masked signal is then fed into the second circuit which synchronizes itself with the chaotic noise.  The output of the second circuit can then be used to subtract off the noise in the masked signal resulting in the original signal.  In practice, the final decoded signal has a fair amount of static but the original signal is clearly audible.  The following mp3 file demonstrates the results.  You hear the original signal followed by the masked signal followed by the decoded signal.  In this example, only the chaotic signal was used to filter the masked signal.  In practice, the decoded signal could easily be improved with further filtering.

 ChaoticFilter.mp3