Quotients of Dynamical Systems and Chaos on the Cantor Fan

Let ( X , f ) be a dynamical system. Using an equivalence relation ∼ on X , we introduce the quotient ( X / ∼ , f ⋆ ) of the dynamical system ( X , f ) . In the first part of the paper, we give new results about sensitive dependence on initial conditions of ( X / ∼ , f ⋆ ) , transitivity of ( X / ∼...

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Bibliographic Details
Published in:Journal of dynamical and control systems 2024-09, Vol.30 (3), Article 33
Main Authors: Banič, Iztok, Erceg, Goran, Kennedy, Judy, Nall, Van
Format: Article
Language:English
Subjects:
Calculus of Variations and Optimal Control
Optimization
Control
Dynamical Systems
Dynamical Systems and Ergodic Theory
Initial conditions
Mathematics
Mathematics and Statistics
Quotients
Systems Theory
Vibration
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Summary:Let ( X , f ) be a dynamical system. Using an equivalence relation ∼ on X , we introduce the quotient ( X / ∼ , f ⋆ ) of the dynamical system ( X , f ) . In the first part of the paper, we give new results about sensitive dependence on initial conditions of ( X / ∼ , f ⋆ ) , transitivity of ( X / ∼ , f ⋆ ) , and periodic points in ( X / ∼ , f ⋆ ) . In the second part of the paper, we use these results to study chaotic functions on the Cantor fan. Explicitly, we study functions f on the Cantor fan C such that (1) ( C , f ) is chaotic in the sense of Devaney, (2) ( C , f ) is chaotic in the sense of Robinson but not in the sense of Devaney, and (3) ( C , f ) is chaotic in the sense of Knudsen but not in the sense of Devaney. We also study chaos on the Lelek fan.
ISSN:1079-2724
1573-8698
DOI:10.1007/s10883-024-09708-x