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Dynamical entropy of the separatrix map
1Russian Academy of Sciences, Institute of Applied Astronomy, Saint Petersburg State University, 7/9 Universitetskaya nab., 199034 Saint Petersburg, Russia and , of the , 191187 Saint Petersburg, Russia.
Physical Review. E
|April 18, 2025
Summary
This study analyzes the chaotic layer in the separatrix map, finding that dynamical entropy (h) is nearly linear with the adiabaticity parameter (λ). This provides a basis for calculating entropy in nonlinear resonance systems.
Area of Science:
- Nonlinear Dynamics
- Chaos Theory
- Statistical Mechanics
Background:
- The behavior of dynamical systems near separatrices is crucial for understanding chaos.
- The separatrix map is a key model for studying chaotic layers in Hamiltonian systems.
- Characterizing the chaotic layer's properties, like its width and Lyapunov exponent, is essential.
Purpose of the Study:
- To calculate the maximum Lyapunov exponent and chaotic layer width as functions of the adiabaticity parameter (λ).
- To investigate the behavior of dynamical entropy (h) in relation to the adiabaticity parameter.
- To establish a foundation for calculating dynamical entropy in perturbed nonlinear resonance systems.
Main Methods:
- Calculated maximum Lyapunov exponent and chaotic layer width using the separatrix map in natural variables.
- Considered the least perturbed border of the chaotic layer, corresponding to the golden mean winding number.
- Analyzed the dependence of dynamical entropy on the adiabaticity parameter.
Main Results:
- The maximum Lyapunov exponent and layer width show strong non-monotonic dependence on the adiabaticity parameter (λ).
- The dynamical entropy (h) exhibits a near-linear relationship with the adiabaticity parameter (λ).
- Normalized dynamical entropy (h/λ) is found to be a quasi-constant.
Conclusions:
- The dynamical entropy of the separatrix map's chaotic layer is approximately linear with the adiabaticity parameter.
- This linear relationship offers a simplified approach to estimating dynamical entropy in perturbed nonlinear resonances.
- Further investigation into the exact linearity of h(λ) is warranted.
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