Computation of entropy and Lyapunov exponent by a shift transform

Area of Science:

• Nonlinear dynamics
• Computational mathematics
• Chaos theory

Background:

• Estimating topological entropy and Lyapunov exponents is crucial for understanding nonlinear dynamical systems.
• Traditional methods often require advanced computational or mathematical techniques, limiting accessibility.
• Identifying measure-zero non-chaotic regions (windows) within chaotic systems is particularly challenging.

Purpose of the Study:

• To introduce a novel, simplified computational method for calculating topological entropy and Lyapunov exponents.
• To demonstrate the method's ability to accurately analyze both chaotic and non-chaotic regions in nonlinear maps.
• To show the equivalence between Kolmogorov-Sinai entropy and topological entropy for the physical measure.

Main Methods:

• A shift transform-based computational approach is utilized.
• The method avoids complex techniques like periodic orbit computation or Markov partitions.
• It is applied to nonlinear maps to estimate key dynamical quantities.

Main Results:

• The method accurately estimates topological entropy and Lyapunov exponents.
• It successfully captures both chaotic dynamics and elusive non-chaotic windows.
• The study confirms that Kolmogorov-Sinai entropy equals topological entropy for the Sinai-Ruelle-Bowen measure.

Conclusions:

• The proposed shift transform method offers a computationally accessible way to analyze nonlinear maps.
• This technique enhances the study of chaotic systems by including important non-chaotic phenomena.
• The findings contribute to a deeper understanding of entropy in dynamical systems.