Chaotic source separation solved by a tank of water through invertible generalized synchronization*

Statistical methods such as independent component analysis and principal component analysis were proposed to separate a set of source signals from a mixed signal by assuming the statistical independence of sources. Here, we focus on chaotic source separation where the sources are chaotic systems. We assume that one has no knowledge about the governing equations of the source signals, and that the mixed signal is simply the sum of the sources. From the perspective of dynamical systems, we propose a supervised learning framework that can solve this problem through an intermediate dynamical system. To demonstrate the power of this framework, we employ a simulated tank of water as the intermediate system, and train it to regenerate chaotic signals from a mixed signal that is the sum of any 2 chaotic trajectories from 6 chaotic systems. We elucidate the underlying mechanism as constructing a nonlinear state-observer utilizing the concept of invertible generalized synchronization. We predict that it is impossible to perfectly separate the chaotic sources if the two sources systems are governed by the same dynamical equations.