Analytic signals and the transition to chaos in deterministic flows

The transition from regular to chaotic motions in deterministic flows is characterized by a change from a discrete Fourier spectrum to a broadband one. The onset of chaos is thus associated with the creation of an infinite number of new Fourier modes. Given a system that generates a time series [Formula Presented] we study the transition to chaos from the perspective of analytic signals, which are defined via the Hilbert transform. In order to identify distinct analytic signals, we decompose the original time series [Formula Presented] into a finite number of modes that correspond to proper rotations in the complex plane of their analytic signals. We provide numerical evidence that at the transition, there is no substantial change in the number of analytic signals characterizing [Formula Presented] Furthermore, the distributions of the instantaneous frequencies of the analytic signals in the chaotic regime are well localized and exhibit no broadband feature. These results suggest a simple organization of chaos in terms of analytic signals.