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Relationships among Various Chaos for Linear Semiflows Indexed with Complex Sectors

Author

Listed:
  • 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)

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 + .

Suggested Citation

  • Shengnan He & Xin Liu & Zongbin Yin & Xiaoli Sun, 2024. "Relationships among Various Chaos for Linear Semiflows Indexed with Complex Sectors," Mathematics, MDPI, vol. 12(20), pages 1-10, October.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:20:p:3167-:d:1495788
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