Using Chaos to Send Secret Messages

One of the most exciting recent developments in nonlinear dynamics is the realization that chaos can be useful. Normally one thinks of chaos as a fascinating curiosity at best, and a nuisance at worst, something to be avoided or engineered away. But since about 1990, people have found ways to exploit chaos to do some marvelous and practical things. For an introduction to this new subject, see Vohra et al. (1992). 

Kevin Cuomo and Alan Oppenheim (1992, 1993) have implemented a new approach to this problem, building on Pecora and Carroll s (1990) discovery of synchronized chaos. Here s the strategy When you transmit the message to your friend, you also mask it with much louder chaos. An outside listener only hears the chaos, which sounds like meaningless noise. But now suppose that your friend has a magic receiver that perfectly reproduces the chaos—then he can subtract off the chaotic mask and listen to the message  

The voltages u,v,w at three different points in the circuit are proportional to Lorenz s x, y,z. Thus the circuit acts like an analog computer for the Lorenz equations. Oscilloscope traces of u t) vs. w t), for example, confirmed that the circuit was following the familiar Lorenz attractor. Then, by hooking up the circuit to a loudspeaker, Cuomo enabled us to hear the chaos—it sounds like static on the radio. 

The astonishing result is that the receiver asymptotically approaches perfect synchrony with the transmitter, starting from any initial conditions. To be precise, let 

Why is this astonishing Because at each instant the receiver has only partial 

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A unique feature of the book is its emphasis on applications. These include mechanical vibrations, lasers, biological rhythms, superconducting circuits, insect outbreaks, chemical oscillators, genetic control systems, chaotic waterwheels, and even a technique for using chaos to send secret messages. In each case, the scientific background is explained at an elementary level and closely integrated with the mathematical theory. 

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