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 into a linear time-invariant form via the 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. First, a Guyan linear reduction method is applied to reduce the order. The second method is to determine 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 nondominant system states. Furthermore, in this work, all three order reduction techniques are demonstrated on the class of commutative and noncommutative/Hills-type nonlinear quasi-periodic systems. The behavior of the reduced system states of the resulting solution is compared with the numerical integration results and their performance is studied using the error plots for each technique.