On Computation of Approximate Lyapunov-Perron Transformations

Abstract

Abstract Many dynamical systems can be modeled by a set of linear/nonlinear ordinary differential equations with quasi-periodic coefficients. Application of Lyapunov-Perron (L-P) transformations to such systems produce dynamically equivalent systems in which the linear parts are time-invariant. In this work, a technique for the computation of approximate L-P transformations is suggested. First, a quasi-periodic system is replaced by a periodic system with a ‘suitable’ large principal period to which Floquet theory can be applied. Then, the state transition matrix (STM) of the periodic system is computed in the symbolic form using shifted Chebyshev polynomials and Picard iteration method. Finally, since the STM can be expressed in terms of a periodic matrix and a time-invariant matrix (Lyapunov-Floquet theorem), this factorization is utilized to compute approximate L-P transformations. A two-frequency quasi-periodic system is investigated using the proposed method and approximate L-P transformations are generated for stable, unstable and critical cases. These transformations are also inverted by defining the adjoint system to the periodic system. Unlike perturbation and averaging, the proposed technique is not restricted by the existence of a generating solution and a small parameter. Approximate L-P transformations can be utilized to design controllers using time-invariant methods and may also serve as a powerful tool in bifurcation studies of nonlinear quasi-periodic systems.