Dynamic Fourier Series Decomposition with Pools of Strongly Coupled Adaptive Frequency Oscillators

Oscillators model are interesting models for applications that involve synchronization phenomena and are increasingly used in science and engineering. However they have two main limitations, first their synchronization properties are limited in the sense that they have a finite entrainment basin, second they have no memory of past interactions (i.e. they come back to their intrinsic frequency whenever the entraining signal disappears). We recently proposed a general mechanism to transform an oscillator into an adaptive frequency oscillator, i.e. an oscillator that can adapt its parameters to learn the frequency of any input signal. This mechanism is such that the entrainment basin becomes infinite and that the oscillator remembers the frequency of entrainment even if the driving signal disappears. Moreover it is generic enough to be applied to a large class of oscillators. In this contribution we detail the fundamental properties of this mechanism in the case of phase oscillators. We show that the frequency adaptation is exponential in the strong coupling case and we generalize the mechanism to be able to explicitly control the relaxation time during convergence. We also show the limits of the system in terms of time-frequency resolution and relate this to an equivalent of Heisenberg boxes. Finally we use these results to extend our previous work on pools of adaptive frequency oscillators, where a set of oscillators coupled via a negative mean field can perform a kind of windowed Fourier series decomposition from a dynamical systems point of view. We augment the system such that the energy content of each frequency component can also be learned and use our results on single oscillators to infer the basic properties of the system. We also show several numerical simulations to illustrate the capabilities of the system.