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Devaney chaos in a Two-Dimensional Space with Weak Topology

  • Hongbo Zeng
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Abstract

It is well known that a finite-dimensional linear system cannot be chaotic. In this article, it shows that Devaney chaos with the weak topology can be generated by a linear map, where the weak topology into a two-dimensional Euclidean space is induced by a linear functional. Especially, it gives the equivalent conditions to be strongly transitive (weakly transitive, strongly sensitive,weakly sensitive, dense periodic points, respectively). Besides, we show that for a matrix with two conjugate complex eigenvalues, strong transitivity (sensitivity, respectively) is equivent to weak transitivity (sensitivity, respectively), which does not hold if we consider a matrix with two real eigenvalues. Finally, in two-dimensional space with weak topology, we show that weak transitivity and weak density of periodic points do not imply weak sensitivity and that weak transitivity and fixed point do not imply Li-Yorke chao, which show that there is a significant difference between the dynamic properties of weak topology and the dynamic properties of norm topology.

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Cite

BibTeXRIS
BibTeX
@misc{zeng2026devaney,
  title = {{Devaney chaos in a Two-Dimensional Space with Weak Topology}},
  author = {Hongbo Zeng},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.27615},
  eprint = {2608.27615},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.27615},
}
RIS
TY  - UNPB
TI  - Devaney chaos in a Two-Dimensional Space with Weak Topology
AU  - Hongbo Zeng
PY  - 2026
DA  - 2026/08/27
PB  - arXiv
DO  - 10.48550/arxiv.2608.27615
UR  - https://arxiv.org/abs/2608.27615
AB  - It is well known that a finite-dimensional linear system cannot be chaotic. In this article, it shows that Devaney chaos with the weak topology can be generated by a linear map, where the weak topology into a two-dimensional Euclidean space is induced by a linear functional. Especially, it gives the equivalent conditions to be strongly transitive (weakly transitive, strongly sensitive,weakly sensitive, dense periodic points, respectively). Besides, we show that for a matrix with two conjugate complex eigenvalues, strong transitivity (sensitivity, respectively) is equivent to weak transitivity (sensitivity, respectively), which does not hold if we consider a matrix with two real eigenvalues. Finally, in two-dimensional space with weak topology, we show that weak transitivity and weak density of periodic points do not imply weak sensitivity and that weak transitivity and fixed point do not imply Li-Yorke chao, which show that there is a significant difference between the dynamic properties of weak topology and the dynamic properties of norm topology.
ER  -