Multiscale measures of phase-space trajectories.

Area of Science:

• Nonlinear Dynamics
• Time Series Analysis
• Complex Systems

Background:

• Characterizing multiscale fluctuations in nonlinear and nonstationary time series is a key challenge in nonlinear sciences.
• Existing methods often struggle to capture the complex, scale-dependent behavior inherent in such data.

Purpose of the Study:

• To develop a unified analysis framework for characterizing multiscale fluctuations in nonlinear and nonstationary time series.
• To integrate empirical mode decomposition (EMD) with generalized fractal dimensions for a comprehensive approach.

Main Methods:

• The study combines empirical mode decomposition (EMD) to derive intrinsic mode functions (IMFs) representing local scale information.
• Generalized fractal dimensions are then calculated from these IMFs to provide multiscale measures of system dynamics.
• The framework is tested on diverse systems, including low- and high-dimensional deterministic models, fractional Brownian motion, and real-world time series.

Main Results:

• The unified EMD-fractal dimension framework successfully characterizes multiscale properties across various dynamical systems.
• The method demonstrates robustness against factors like noise, initial conditions, and time series length.
• Application to paleoclimate and geomagnetic data confirms its utility for real-world time series analysis.

Conclusions:

• The proposed formalism offers a powerful tool for dissecting the multiscale nature of fluctuations in complex systems.
• This integrated approach enhances the understanding of nonlinear dynamics and nonstationarity in diverse scientific fields.
• The framework's applicability to both simulated and real-world data highlights its broad potential.