Stability and reducibility of quasi-periodic systems

In this work, we focused on the stability and reducibility of quasi-periodic systems. We examined the quasi-periodic linear Mathieu equation of the form x ̈+(ä+ϵ[cost+cosùt])x=0 The stability of solutions of Mathieu's equation as a function of parameter values (ä,ϵ) had been analyzed in this work. We used the Floquet type theory to generate stability diagrams which were used to determine the bounded regions of stability in the ä-ù plane for fixed ϵ. In the case of reducibility, we first applied the Lyapunov- Floquet (LF) transformation and modal transformation, which converted the linear part of the system into the Jordan form. Very importantly, quasi-periodic near-identity transformation was applied to reduce the system equations to a constant coefficient system by solving homological equations via harmonic balance. In this process we obtained the reducibility/resonance conditions that needed to be satisfied to convert a quasi-periodic system to a constant one.