Using the machine learning approach known as reservoir computing, it is possible to train one dynamical system to emulate another. We show that such trained reservoir computers reproduce the properties of the attractor of the chaotic system sufficiently well to exhibit chaos synchronization. That is, the trained reservoir computer, weakly driven by the chaotic system, will synchronize with the chaotic system. Conversely, the chaotic system, weakly driven by a trained reservoir computer, will synchronize with the reservoir computer. We illustrate this behavior on the Mackey-Glass and Lorenz systems. We then show that trained reservoir computers can be used to crack chaos based cryptography and illustrate this on a chaos cryptosystem based on the Mackey-Glass system. We conclude by discussing why reservoir computers are so good at emulating chaotic systems.

The Koopman operator provides a powerful framework for data-driven analysis of dynamical systems. In the last few years, a wealth of numerical methods providing finite-dimensional approximations of the operator have been proposed [e.g., extended dynamic mode decomposition (EDMD) and its variants]. While convergence results for EDMD require an infinite number of dictionary elements, recent studies have shown that only a few dictionary elements can yield an efficient approximation of the Koopman operator, provided that they are well-chosen through a proper training process. However, this training process typically relies on nonlinear optimization techniques. In this paper, we propose two novel methods based on a reservoir computer to train the dictionary. These methods rely solely on linear convex optimization. We illustrate the efficiency of the method with several numerical examples in the context of data reconstruction, prediction, and computation of the Koopman operator spectrum. These results pave the way for the use of the reservoir computer in the Koopman operator framework.

Reservoir computing is a machine learning approach to designing artificial neural networks. Despite the significant simplification of the training process, the performance of such systems is comparable to other digital algorithms on a series of benchmark tasks. Recent investigations have demonstrated the possibility of performing long-horizon predictions of chaotic systems using reservoir computing. In this work we show that a trained reservoir computer can reproduce sufficiently well the properties a chaotic system, hence allowing full synchronisation. We illustrate this behaviour on the Mackey-Glass and Lorenz systems. Furthermore, we show that a reservoir computer can be used to crack chaos-based cryptographic protocols and illustrate this on two encryption schemes.

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