Estimating Phase From Observed Trajectories Using the Temporal 1-Form

Area of Science:

• Dynamical Systems
• Nonlinear Dynamics
• Time Series Analysis

Background:

• Oscillators are fundamental in nature, with dynamics often dictated by an asymptotic phase.
• Existing data-driven methods for phase estimation can be limited by data length and noise sensitivity.

Purpose of the Study:

• To develop a robust algorithm for estimating the asymptotic phase of oscillators.
• To enable accurate recovery of phase response curves (PRCs) and isochron geometry from time-series data.

Main Methods:

• A novel series expansion is employed to directly compute the phase response curve (PRC).
• An algorithm is provided for estimating the coefficients of this series expansion.
• The method is designed to handle short observations and varying noise conditions.

Main Results:

• The algorithm accurately estimates asymptotic phase even with data shorter than one cycle.
• Proven convergence rate bounds are established concerning measurement and system noise.
• The method successfully recovers PRCs, isochron curvature, and nonlinear geometric features.

Conclusions:

• This data-driven approach offers a powerful tool for modeling oscillator dynamics from observed time-series.
• The algorithm's ability to use short data segments and its noise resilience make it broadly applicable.
• Potential applications include constructing oscillator models in diverse scientific and engineering fields.