Entropy-based generating Markov partitions for complex systems

Area of Science:

• Dynamical Systems and Network Science
• Information Theory
• Computational Physics

Background:

• Encoding dynamical system trajectories into symbolic sequences is challenging due to the need to preserve invariant properties.
• Generating Markov Partitions (GMPs) theoretically solve this but require infinite precision knowledge of system dynamics, which is often unattainable in real-world experiments with finite resolution and time spans.
• Analyzing high-dimensional complex systems, like networks of interacting units, further complicates trajectory encoding.

Purpose of the Study:

• To develop a method for the approximate construction of Generating Markov Partitions (GMPs) for complex systems using finite-resolution and finite-time trajectory data.
• To encode trajectories of complex systems into optimal symbolic sequences that minimize information loss and spurious information.
• To enable the calculation of invariant probability measures and complexity measures from observed data in complex systems.

Main Methods:

• Developed a novel method to approximate Generating Markov Partitions (GMPs) from finite-resolution and finite-time trajectories of complex systems.
• Applied the method to networks of coupled maps, generating symbolic sequences from their trajectories.
• Validated the optimality of the generated symbolic sequences by demonstrating minimized information loss and spurious information.

Main Results:

• Successfully constructed approximate GMPs for complex systems from experimental data.
• Generated symbolic sequences that are optimal in terms of information preservation.
• Demonstrated the ability to calculate invariant probability measures of complex systems using the derived symbolic sequences.

Conclusions:

• The developed method provides an effective way to encode complex system dynamics into symbolic sequences, overcoming limitations of finite data resolution and time.
• This approach facilitates the accurate calculation of invariant probability measures and the definition of complexity measures for diverse complex phenomena.
• The method has potential applications in analyzing real-world data, such as electroencephalogram signals for brain activity and temperature anomalies for climate variability.