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An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids
Published on: December 4, 2017
The experimental localization of Aubry-Mather sets using regularization techniques inspired by viscosity theory.
Massimiliano Guzzo1, Olga Bernardi, Franco Cardin
1Dipartimento di Matematica Pura ed Applicata, Università degli Studi di Padova, via Trieste 63, 35121 Padova, Italy.
Chaos (Woodbury, N.Y.)
|October 2, 2007
Summary
We introduce a novel method using "relative viscosity and friction" to locate Aubry-Mather sets in dynamical systems. This technique regularizes invariant sets, enabling precise mapping and gap identification in phase-space curves.
Area of Science:
- Dynamical Systems and Chaos Theory
- Mathematical Physics
Background:
- Aubry-Mather sets are crucial for understanding the dynamics of quasi-integrable systems.
- Locating these sets, especially those with irrational rotation numbers, presents significant computational challenges.
Purpose of the Study:
- To develop a new, robust method for the precise localization of Aubry-Mather sets.
- To introduce and apply the concept of "relative viscosity and friction" for regularization techniques.
Main Methods:
- Development of regularization techniques inspired by viscosity theories.
- Introduction of "relative viscosity and friction" for parameterizing invariant sets.
- Computation of phase-space curves and invariant measures to identify set locations and gaps.
Main Results:
- Successfully regularized parametrizations of invariant sets with irrational rotation numbers.
- Developed a method to compute curves near Aubry-Mather sets and invariant measures.
- Demonstrated the method's application to the standard map's golden cantorus and a general case.
Conclusions:
- The proposed method effectively overcomes challenges in localizing Aubry-Mather sets.
- The concept of relative viscosity and friction offers a powerful tool for analyzing complex dynamical systems.
- This approach advances the study of invariant sets in two-dimensional twist maps.
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