A Floquet–Lyapunov theory for nonautonomous linear periodic differential equations with piecewise constant deviating arguments

Abstract

In this work, we introduce a version of the classic Floquet–Lyapunov theorem for $\omega -$periodic nonautonomous linear differential equations with piecewise constant arguments of generalized type (IDEPCAG or DEPCAG). We demonstrate that the nonautonomous linear IDEPCAG is kinematically similar to an autonomous linear ordinary differential equation. Additionally, we provide explicit formulas for the Floquet normal form of the fundamental matrix of IDEPCAG nonautonomous linear systems. These are very useful for analyzing qualitative properties such as the stability and periodicity of the solutions, making the study of IDEPCAG systems more accessible. Finally, we have included examples to demonstrate the effectiveness of our results.