Relationships among Various Chaos for Linear Semiflows Indexed with Complex Sectors

Shengnan He (), Xin Liu, Zongbin Yin and Xiaoli Sun
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Shengnan He: School of Humanities and Fundamental Sciences, Shenzhen Institute of Information Technology, Shenzhen 518172, China
Xin Liu: School of Mathematics, Guangdong University of Education, Guangzhou 510303, China
Zongbin Yin: School of Mathematics and Systems Science, Guangdong Polytechnic Normal University, Guangzhou 510665, China
Xiaoli Sun: School of Mathematical Sciences, Shenzhen University, Shenzhen 518172, China

Mathematics, 2024, vol. 12, issue 20, 1-10

Abstract: In this paper, we investigate the relationships among point transitivity, topological transitivity, Li–Yorke chaos, and the existence of irregular vectors for a linear semiflow { T t } t ∈ Δ indexed with a complex sector. We reveal the equivalence between topological transitivity and point transitivity for a linear semiflow { T t } t ∈ Δ , especially in case the range of some operator T t , t ∈ Δ is not dense. We also prove that Li–Yorke chaos is equivalent to the existence of a semi-irregular vector and that point transitivity is stronger than the existence of an irregular vector for any linear semiflow T t t ∈ Δ . At last, unlike the conclusion for traditional linear dynamical systems, we show that there exists a Li–Yorke chaotic C 0 -semigroup T t t ∈ Δ without irregular vectors. The results and proof methods presented in this paper demonstrate the differences in the dynamical behavior between linear semiflows { T t } t ∈ Δ and traditional linear systems with the acting semigroup S = Z + and S = R + .

Keywords: linear semiflows; complex sector; transitivity; Li–Yorke chaos; irregular vector (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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