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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
PubMed
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.

Keywords:
Lorenz mapcompletely invariant setentropylimit setrenormalizable map

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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.