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On α-Limit Sets in Lorenz Maps
Łukasz Cholewa1, Piotr Oprocha1,2
1Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Kraków, Poland.
Entropy (Basel, Switzerland)
|September 28, 2021
Summary
This study reveals that alpha-limit sets in Lorenz maps can exhibit unexpected dynamics, not always being completely invariant. This finding also extends to unimodal maps and connects to topological entropy calculations.
Area of Science:
- Dynamical Systems
- Chaos Theory
- Mathematical Analysis
Background:
- Dynamical systems theory studies the long-term behavior of systems.
- Lorenz maps and unimodal maps are key models in this field.
- Understanding invariant sets is crucial for predicting system behavior.
Purpose of the Study:
- To demonstrate that alpha-limit sets in Lorenz maps are not necessarily completely invariant.
- To identify and analyze unexpected dynamical behaviors in these maps.
- To explore the relationship between alpha-limit sets and topological entropy.
Main Methods:
- Analysis of Lorenz maps.
- Investigation of unimodal maps on the interval [0,1].
- Study of the structure of alpha-limit sets.
- Calculation of topological entropy.
Main Results:
- Provided examples show alpha-limit sets in Lorenz maps are not always completely invariant.
- Similar behavior is observed in unimodal maps.
- The structure of alpha-limit sets is directly linked to topological entropy calculation.
Conclusions:
- The research fills a gap in the literature regarding the invariance of alpha-limit sets.
- The findings highlight novel dynamical behaviors in Lorenz and unimodal maps.
- A clear connection is established between the structure of alpha-limit sets and topological entropy.
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