Approximate Lyapunov–Perron Transformations: Computation and Applications to Quasi-Periodic Systems

Abstract

A new technique for the analysis of dynamical equations with quasi-periodic coefficients (so-called quasi-periodic systems) is presented. The technique utilizes Lyapunov–Perron (L–P) transformation to reduce the linear part of a quasi-periodic system into the time-invariant form. A general approach for the construction of L–P transformations in the approximate form is suggested. First, the linear part of a quasi-periodic system is replaced by a periodic system with a “suitable” large principal period. Then, the state transition matrix (STM) of the periodic system is computed in the symbolic form using Floquet theory. Finally, Lyapunov–Floquet theorem is used to compute approximate L–P transformations. A two-frequency quasi-periodic system is studied and transformations are generated for stable, unstable, and critical cases. The effectiveness of these transformations is demonstrated by investigating three distinct quasi-periodic systems. They are applied to a forced linear quasi-periodic system to generate analytical solutions. It is found that the closeness of the analytical solutions to the exact solutions depends on the principal period of the periodic system. A general approach to obtain the stability bounds on linear quasi-periodic systems with stochastic perturbations is also discussed. Finally, the usefulness of approximate L–P transformations is presented by analyzing a nonlinear quasi-periodic system with cubic nonlinearity using time-dependent normal form (TDNF) theory. The closed-form solution generated is found to be in good agreement with the exact solution.