Some properties of the Persidskiǐ spectrum (Q2387936)

Consider the system of linear differential equations \[ x'=A(t)x,\tag{1} \] where the coefficients of \(A(t)\) are continuous and bounded on \([0, \infty)\). Let \(\{t_k\} \subset(0,\infty)\), \(\lim_{k\to\infty}t_k=\infty\) and let \(A_k(t)\) be the \(t_k\)-periodic matrix function such that \(A(t)= A_k(t)\) for \(t\in[0,t_k)\). Let \((\lambda_1^{(k)},\dots,\lambda_n^{(k)})\) be the complete spectrum of the system \[ x'=A_k(t)x,\tag{2} \] \(\lambda_1^{(k)} \leq\lambda_1^{(k)} \leq\cdots\leq \lambda_n^{(k)}\). The Persidskii spectrum \({\mathcal P}\) is the set of all \(\lambda^*= (\lambda^*_1,\dots,\lambda^*_n)\in\mathbb{R}^n\) for which there exists a sequence \(\{t_k\}\subset (0,\infty)\), \(\lim_{k\to\infty}t_k=\infty\) such that \(\lim_{k\to\infty}\lambda_i^{(k)}=\lambda^*_i\) for \(i=1,\dots,n\). In this paper, the author shows that the Persidskii spectrum is a compact connected set and discusses in detail its structure for proper second-order systems. He also shows on examples of diagonal two-dimensional systems that the properties of the Persidskii spectrum can be very different for improper systems.