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).
The basic idea is as follows. First, we construct an electronic circuit
using operational amplifiers configured as integrators to replicate the
Lorenz Equations. The pictures below shows oscilloscope screenshots when
viewing the different outputs in x-y mode.
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 below shows an oscilloscope
screenshot when viewing the outputs of the two circuits in x-y mode.
Lastly, these synchronized circuits can be used to send "secret" messages.
The first circuit is used to encode an audio signal (music, for example) with
high amplitude chaotic noise. This 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 original 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 three wave files demonstrate the results.
Original Signal
Masked Signal
Decoded Signal