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Using the machine learning approach known as reservoir computing, it ispossible to train one dynamical system to emulate another. We show that suchtrained reservoir computers reproduce the properties of the attractor of thechaotic system sufficiently well to exhibit chaos synchronisation. That is, thetrained reservoir computer, weakly driven by the chaotic system, willsynchronise with the chaotic system. Conversely, the chaotic system, weaklydriven by a trained reservoir computer, will synchronise with the reservoircomputer. We illustrate this behaviour on the Mackey-Glass and Lorenz systems.We then show that trained reservoir computers can be used to crack chaos basedcryptography and illustrate this on a chaos cryptosystem based on theMackey-Glass system. We conclude by discussing why reservoir computers are sogood at emulating chaotic systems.

Using a reservoir computer to learn chaotic attractors, with applications to chaos synchronisation and cryptography