A Criterion for Distinguishing Temporally Different Dynamical Systems.

Area of Science:

• Dynamical Systems Theory
• Statistical Mechanics
• Information Theory

Background:

• Distinguishing behaviors of dynamical systems is crucial for understanding complex phenomena.
• Existing methods often rely on external reference frames, which may not capture intrinsic system evolution.
• Ergodic dynamical systems possess unique temporal characteristics.

Purpose of the Study:

• To propose and define a novel criterion for differentiating dynamical systems based on their intrinsic behavior.
• To introduce the concept of 'internal time' as a system-generated temporal measure.
• To explore the properties and applications of internal time in dynamical systems analysis.

Main Methods:

• Formal definition of internal time using the entropy ratio.
• Analysis of the fundamental properties of the defined internal time.
• Application of the internal time concept to specific examples of dynamical systems.

Main Results:

• The proposed criterion, interpreted as internal time, effectively distinguishes dynamical systems by their behavior.
• Internal time is formally defined as the entropy ratio of an ergodic dynamical system.
• Examples demonstrate the utility of internal time in analyzing system dynamics.

Conclusions:

• Internal time offers a new perspective for characterizing dynamical systems, independent of external time.
• The entropy ratio provides a quantifiable measure for this intrinsic temporal property.
• This framework enhances the analysis and differentiation of complex system behaviors.