Decomposing Signals from Dynamical Systems Using Shadow Manifold Interpolation

Erin George, Colleen E. Chan, Gal Dimand, Ryan M. Chakmak, Claudia Falcon, Daniel Eckhardt, Robert Scott Martin

Abstract: For signals driven by multiple separate and decoupled dynamical systems, traditional methods in signal analysis often fail to decompose the signal into the parts driven by the different dynamics. Two signals from different dynamical systems may have similar energy at every frequency, so techniques such as the Fourier transform will be unable to separate a time series if it contains multiple such signals. This is especially true when the dynamical system is chaotic, as the produced signals are inherently broadband. We present how an algorithm called shadow manifold interpolation (SMI), inspired by recent advances in applied dynamical systems theory, can succeed in this task. SMI takes two causally related signals and reconstructs one from the other, producing a reconstruction that only captures the shared dynamics between the two signals, i.e., the portion of the signals driven by some underlying dynamical systems that contribute to both signals. From this we produce a decomposition of the signal into different signals representing separate dynamics. Furthermore, we demonstrate the effectiveness of SMI at decomposing signals in a variety of test cases. We consider three ways in which two signals may be causally related. Through testing the algorithm on simulated composite dynamical systems, we demonstrate that SMI succeeds in separating out the constituent systems in two schemes, failing only when the two signals share multiple decoupled dynamics. The main limitation of this algorithm is that a second causally related reference signal is needed in order to decompose a given signal. This reference signal must share at least one dynamic of interest with the given signal.