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Topological characterization of deterministic chaos: enforcing orientation preservation
Marc Lefranc1, Pierre-Emmanuel Morant, Michel Nizette
1Laboratoire de Physique des Lasers, Atomes et Molécules, UMR CNRS 8523 Centre d'Etudes et de Recherches Lasers et Applications, Université des Sciences et Technologies de Lille, Cedex, France. marc.lefranc@univ-lille.fr
Summary
We introduce a new framework to study chaos theory and nonlinear dynamics. This method enforces the determinism principle, enabling topological analysis of complex systems in any dimension.
Area of Science:
- Nonlinear dynamics
- Chaos theory
- Topology
Background:
- The determinism principle is fundamental to nonlinear dynamics and chaos theory.
- It prevents state space trajectories from intersecting, crucial for topological analysis of chaos.
- Knot theory, used for this analysis, is limited to three-dimensional systems.
Purpose of the Study:
- To propose an alternative framework for analyzing chaos that applies to systems of any dimension.
- To enforce the determinism principle within this new framework.
- To validate the approach by comparing predictions with known results.
Main Methods:
- Constructing an orientation-preserving dynamics on triangulated surfaces.
- Applying the framework to analyze the topological properties of unstable periodic orbits.
- Numerically predicting topological entropies for periodic orbits.
Main Results:
- The proposed framework successfully enforces the determinism principle.
- In three dimensions, the method accurately predicts topological entropies for the horseshoe map's periodic orbits.
- Demonstrates a viable approach for topological analysis beyond three-dimensional systems.
Conclusions:
- The developed framework offers a generalized method for topological analysis in nonlinear dynamics and chaos theory.
- It overcomes the dimensional limitations of traditional knot-theoretic approaches.
- This work provides a new perspective on understanding deterministic chaos across different system dimensions.
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