Lyapunov-Perron Transformation for Quasi-Periodic Systems and Its Applications

Susheelkumar C. Subramanian, Sangram Redkar

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


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.

Original languageEnglish (US)
Article number041015
JournalJournal of Vibration and Acoustics, Transactions of the ASME
Issue number4
StatePublished - Aug 2021


  • Dynamics
  • Mechatronics and electro-mechanical systems
  • Nonlinear vibration
  • Stability
  • Vibration control

ASJC Scopus subject areas

  • Acoustics and Ultrasonics
  • Mechanics of Materials
  • Mechanical Engineering


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