Covariant influences for finite discrete dynamical systems

Area of Science:

• Dynamical Systems Theory
• Information Theory
• Mathematical Physics

Background:

• External influences on dynamical systems are typically studied as small perturbations.
• Existing theories often fail to account for significant or non-perturbative external influences.
• A unified framework for understanding influences across deterministic and stochastic systems is lacking.

Purpose of the Study:

• To develop a rigorous theory of external influences on finite discrete dynamical systems that accommodates non-perturbative effects.
• To establish a framework for covariant influences using resource theories.
• To identify conditions governing state transitions under deterministic and stochastic covariant influences.

Main Methods:

• Development of a theory for covariant influences based on resource theories, adapted from quantum information theory.
• Formulation of a covariance condition: evolving the system then influencing it is equivalent to influencing then evolving.
• Analysis of state transition conditions for both deterministic and stochastic dynamical systems.

Main Results:

• A general theory of covariant influences applicable to finite discrete dynamical systems.
• Necessary and sufficient conditions for state transitions under deterministic covariant influences.
• Necessary conditions for state transitions under stochastic covariant influences, identifying forbidden transitions.
• Unification of the behavior of diverse finite discrete dynamical systems under a single theoretical framework.

Conclusions:

• The developed theory provides a rigorous and checkable mathematical framework for understanding external influences on dynamical systems.
• The application of resource theories offers a novel perspective for analyzing discrete dynamical systems.
• The findings predict fundamental constraints on state transitions, applicable across various system types.