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

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.