Extending Lyapunov-Perron transformation for the reducibility of quasiperiodic system

The reducibility of quasiperiodic (QP) linear systems via Lyapunov-Perron (LP) transformation plays a pivotal role in Hamiltonian dynamics and may become a powerful tool in QP response analysis. The reducibility has received significant research interest, among which most focused on qualitative analysis of some local properties. The quantification of reducibility has been a challenging topic for decades, especially within a relatively large parametric range. In this paper, we will propose a practical and quantitative approach to assess the reducibility of the QP Hamiltonian system, based on a semi-analytical approach recently established to determine the LP transformation. The critical value of a certain parameter, of which the system can be reduced via routine LP transformation only on one side, can be determined accurately by the proposed approach. On the other side of the critical value, more importantly, an extension is proposed for the LP transformation to reduce the system, whereas the routine transformation ceases to be valid. With the critical value, the parametric space can be divided into regions where the routine LP transformation is applicable or the extended transformation needs to be employed. The extended transformation would be applicable in QP response analysis and could potentially enrich the research on QP system reducibility within a large parametric space.