Li-Yorke chaos of linear differential equations in a finite-dimensional space with a weak topology (Q6550718)

It is well known that linear discrete dynamical systems cannot be chaotic on finite-dimensional spaces. The authors introduce the notion of Li-Yorke chaos of linear differential equations in a finite-dimensional space with a weak topology and show that a flow generated from a linear differential equation can be Li-Yorke chaotic in this sense:\N\NTheorem. Consider a linear system \(\dot{x}=A x, x \in \mathbb{R}^n\), with an \(2 n \times 2 n\) matrix A, and suppose that there exists a non-singular matrix \(S\) such that\N\[\NS^{-1} A S=\operatorname{diag}\left(\left(\begin{array}{cccc} \lambda & 1 & & \\\N& \lambda & \ddots & \\\N& & \ddots & 1 \\\N& & & \lambda \end{array}\right)_{n \times n},\left(\begin{array}{cccc} \bar{\lambda} & 1 & & \\\N& \bar{\lambda} & \ddots & \\\N& & \ddots & 1 \\\N& & & \frac{1}{\lambda} \end{array}\right)_{n \times n}\right)_{2 n \times 2 n} \N\]\Nwhere \(\lambda=\alpha+i \beta\) and \(\bar{\lambda}=\alpha-i \beta\) are complex conjugate eigenvalues, \(\alpha, \beta\) are real numbers, and \(\beta \neq 0\). Then,\N\begin{itemize}\N\item[1.] if \(\alpha<0\), the flow is not Li-Yorke chaotic in weak topology;\N\item[2.] if \(\alpha \geq 0\), the flow is (strongly) Li-Yorke chaotic in weak topology, where the uncountable scrambled set can be taken as any uncountable subset of the whole space.\N\end{itemize}