Lyapunov Perron Transformation for Quasi-Periodic Systems and its Applications

This paper depicts the application of symbolically computed Lyapunov Perron (L-P) Transformation to solve linear and nonlinear quasi-periodic systems. The L-P transformation converts a linear quasi-periodic system into a time-invariant one. State augmentation and the method of Normal Forms are used to compute the L-P transformation analytically. The state augmentation approach converts a linear quasi-periodic system into a nonlinear time invariant system as the quasi-periodic parametric excitation terms are replaced by ‘fictitious’ states. This nonlinear system can be reduced to a linear system via Normal Forms in the absence of resonances. In this process, one obtains near identity transformation that contains fictitious states. Once the quasi-periodic terms replace the fictitious states they represent, the near identity transformation is converted to the L-P transformation. The L-P transformation can be used to solve linear quasi-periodic systems with external excitation and nonlinear quasi-periodic systems. Two examples are included in this work, a commutative quasi-periodic system and a non-commutative Mathieu-Hill type quasi-periodic system. The results obtained via the L-P transformation approach match very well with the numerical integration and analytical results.

Computation of Lyapunov–Perron transformation for linear quasi-periodic systems

The transformation of a linear time periodic system to a time-invariant system is achieved using the Floquet theory. In this work, the authors attempt to extend the same toward the quasi-periodic systems, using a Lyapunov–Perron transformation. Though a technique to obtain the closed-form expression for the Lyapunov–Perron transformation matrix is missing in the literature, the application of unification of multiple theories would aid in identifying such a transformation. In this work, the authors demonstrate a methodology to obtain the closed-form expression for the Lyapunov–Perron transformation analytically for the case of a commutative quasi-periodic system. In addition, for the case of a noncommutative quasi-periodic system, an intuitive state augmentation and normal form techniques are used to reduce the system to a time-invariant form and obtain Lyapunov–Perron transformation. The results are compared with the numerical techniques for validation.