AI 中文总结
本文证明弱拓扑下二维线性映射可产生Devaney混沌，给出相关动力学性质的等价条件，揭示弱拓扑与范数拓扑的动力学性质存在显著差异。
AI 中文摘要
众所周知，有限维线性系统无法产生混沌。本文证明，可通过线性映射在二维欧几里得空间中诱导的弱拓扑下生成Devaney混沌，该弱拓扑由线性泛函诱导。特别地，本文给出了强传递性（分别对应弱传递性、强敏感性、弱敏感性、稠密周期点）的等价条件。此外，本文表明，对于具有两个共轭复特征值的矩阵，强传递性（分别对应敏感性）等价于弱传递性（分别对应敏感性），而若考虑具有两个实特征值的矩阵，该等价性不成立。最后，在弱拓扑下的二维空间中，本文证明弱传递性与周期点的弱稠密性并不蕴含弱敏感性，且弱传递性与不动点并不蕴含Li-Yorke混沌，这表明弱拓扑的动力学性质与范数拓扑的动力学性质存在显著差异。
英文摘要
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.
