Phase definition to assess synchronization quality of nonlinear oscillators

Abstract

This paper proposes a phase definition, named the vector field phase, which can be defined for systems with arbitrary finite dimension and is a monotonically increasing function of time. The proposed definition can properly quantify the dynamics in the flow direction, often associated with the null Lyapunov exponent. Numerical examples that use benchmark periodic and chaotic oscillators are discussed to illustrate some of the main features of the definition, which are that (i) phase information can be obtained either from the vector field or from a time series, (ii) it permits not only detection of phase synchronization but also quantification of it, and (iii) it can be used in the phase synchronization of very different oscillators.

A novel wave decomposition for oscillatory signals

“…For F M M m with m > 2, the conditions for a valid IF are more demanding. Chavez et al (2006) and Freitas et al (2018) show scenarios, like those in Wei and Bovik (1998), where the AS fails, corresponding to signals with more than one dominant oscillation. In this section, two examples are shown that illustrate how AS fails even in scenarios where there is an apparent single oscillation.…”

“…A popular approach, adopted by some authors such as Deng et al (2016), Oprisan (2017), andCaranica et al (2019), is to use the IP associated with the AS approach, defined in (1). The ambiguity on phase definition is well explained in Osipov et al (2003), Chavez et al (2006), andFreitas et al (2018) where other alternatives are also provided.…”

Functional observability and subspace reconstruction in nonlinear systems

“…Oscillations in the slow timescale can be approximately monitored by the "slow phase" variable θ s = x 1 (Fig. 6b), in which a full revolution of the system is completed every time the trajectory approximates the cord filament close to the origin (defining the Poincaré section P = {x : x 1 = 0, ẋ1 > 0}) [40]. Oscillations in the fast timescale, on the other hand, can be monitored by the "fast phase" variable θ f = tan −1 (x 2 /x 3 ) (Fig.…”

“…In practice, even if the original state space is not entirely observable (reconstructible), one may focus on particular subspaces (e.g., state variables) that are relevant to the considered applications. Examples include the estimation of the phase variable of nonlinear oscillators for synchronization analysis of chaotic systems [21,39,40], modeling of climate dynamics [41], and forecasting of financial crashes [42]; the positioning and tracking of a particular spatial coordinate (e.g., altitude) in autonomous aerial vehicles from indirect measurements [43]; or the inference of control variables (which dictate how close a system is to a bifurcation) for the early warning of transitions from healthy to disease states in atrial fibrillation [44] and epileptic seizures [45].…”

Colored Noise in Oscillators. Phase-Amplitude Analysis and a Method to Avoid the itô-Stratonovich Dilemma

“…Since (31)- (32) are exact, they are not easier to solve than the white noise approximated SDE (26). However, they can be used to derive a phase reduced model [26], [29], [30] that in turn can form the basis to find useful, albeit approximate, results. The main advantage of a phase reduced model is that methods for Markovian systems, e.g.…”