The author continues the investigation of symplectic dynamic systems on time scales initiated in his joint paper with the reviewer [O. Došlý and R. Hilscher, J. Difference Equ. Appl. 7, 265-295 (2001; Zbl 0989.34027)].
A symplectic dynamic system on time scales is the linear system \[ z^\Delta=S(t)z, \quad t\in \mathbf T, \tag{*} \] where \(\mathbf T\) is a time scale (i.e., any closed subset of reals \(\mathbf R\)) and \(S\colon \mathbf T\to \mathbf R^{2n\times 2n}\) is a matrix satisfying \(S^T{\mathcal J}+{\mathcal J}S+\mu S^T{\mathcal J}S=0\), \({\mathcal J}=\binom{0\;\;I}{-I\;\;0},\) \(\mu\) being the graininess of the time scale \(\mathbf T\). The main result of the paper is the so-called Reid roundabout theorem for (*), a statement which relates the disconjugacy of (*) in a given time scale interval to the positivity of the corresponding quadratic functional and to the solvability of the associated Riccati matrix equation.
The results of the paper are fundamental in the theory of symplectic dynamic systems and open new areas for the further investigation.