Causality Analysis: Identifying the Leading Element in a Coupled Dynamical System

Area of Science:

• Nonlinear Dynamics
• Complex Systems Analysis
• Coupled Oscillator Systems

Background:

• Physical systems often exhibit time-varying internal couplings, with governing equations frequently unknown.
• Identifying the leading subsystem (largest average coupling coefficient) is crucial for understanding these systems.
• Convergent Cross Mapping (CCM) has previously determined causality in systems with constant couplings.

Purpose of the Study:

• To apply Convergent Cross Mapping (CCM) to coupled Lorenz systems with time-varying coupling coefficients.
• To investigate CCM's ability to identify the dominant subsystem under various coupling schemes and noise conditions.
• To assess CCM's performance when dominant subsystems switch over time.

Main Methods:

• Utilized a pair of coupled Lorenz systems with time-varying coupling coefficients.
• Implemented four sets of numerical experiments with different coupling schemes (Periodic-constant, Normal, Mixed Normal/Non-normal).
• Introduced temporal uncertainties and additive normal noise in experiments to simulate realistic conditions.

Main Results:

• CCM successfully identified the leading subsystem across diverse coupling schemes, including periodic, normal, and mixed variations.
• The method remained effective even when temporal uncertainties and additive noise were imposed.
• CCM failed to clearly distinguish a leading subsystem only when average coupling coefficients were approximately equal.

Conclusions:

• Convergent Cross Mapping (CCM) is a robust method for detecting directional interactions and identifying leading subsystems in complex systems with time-varying couplings.
• CCM's efficacy is demonstrated even under noisy and uncertain conditions, highlighting its practical applicability.
• The study confirms CCM's limitation when distinguishing between subsystems with nearly equivalent average coupling strengths.