Order Reduction of Nonlinear Quasi-periodic Systems Using Lyapunov-perron Transformation

Abstract

In this paper, multiple order reduction techniques for parametrically excited nonlinear quasi-periodic systems are presented. The linear time varying part of the quasi-periodic system is transformed to a Linear Time-Invariant (LTI) form via Lyapunov-Perron (L-P) Transformation. The analytical computation of such a transformation is performed using an intuitive state augmentation and the Normal Forms technique. This L-P transformation is further utilized in analyzing the nonlinear part of the original quasi-periodic system. Using the L-P transformation, three order reduction techniques are detailed in this work. Firstly, a Guyan linear reduction method is applied to reduce the nonlinear terms. The second method is to identify a nonlinear projection based on the Singular Perturbation method. In the third technique, the method of Invariant Manifold is applied to identify a relationship between the dominant and non-dominant system states. Furthermore, in this work, all three order reduction techniques are demonstrated on the class of commutative and non-commutative/Hills type nonlinear quasiperiodic systems. The behavior of the systems states of the resulting solution are compared with the numerical integration results and their performance is rated based on the error plots for each technique.