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.

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