Topological Causality in Dynamical Systems

Area of Science:

• Complex Systems Science
• Nonlinear Dynamics
• Causality Theory

Background:

• Determining causal relations is crucial for understanding complex systems.
• Classical causality measures fail for nonlinear, inseparable systems.
• A transparent measure for causal influences in cyclic systems is needed.

Purpose of the Study:

• Develop a mathematically sound measure for effective causal influences in cyclic deterministic systems.
• Quantify the magnitude and temporal dynamics of causal links.
• Address limitations of classical causality in nonlinear contexts.

Main Methods:

• Utilized time-delay state space reconstructions from observable time series.
• Analyzed expansions of mappings between reconstructed state spaces.
• Defined novel causality indices based on these expansions.

Main Results:

• The method reveals directed coupling strengths and state-dependent influences.
• Novel causality indices accurately capture asymmetry and time-dependence.
• Effective strengths of causal links in complex systems are measurable.

Conclusions:

• The proposed causality indices offer a transparent and effective way to measure causal influences in cyclic systems.
• This approach overcomes limitations of classical causality in nonlinear dynamics.
• Provides a powerful tool for analyzing complex systems across scientific disciplines.