A Direct Approach to Compute the Lyapunov–Perron Transformation for Linear Quasi-periodic Systems

Purpose: As per the dynamical system theory, a Lyapunov–Perron (L–P) transformation can transform a linear quasi-periodic system to a time-invariant form under certain conditions. However, to the best of author’s knowledge, a systematic approach to analytically compute such a transformation is not available in the literature. In this work, a simple yet practical method to compute the L–P transformation matrix is discussed comprehensively. Methods: In this work, the authors demonstrate the conversion of a commutative linear quasi-periodic system into a time-invariant system using Floquet type theory. Moreover, for a linear non-commutative parametrically excited quasi-periodic system satisfying diophantine condition, the authors employ an intuitive state augmentation and the time independent normal forms (TINF) technique to transform it into a time-invariant form. Results: The temporal and phase space variations computed from the proposed approach are compared with the numerical techniques for both commutative and non-commutative quasi-periodic systems. Additionally, the element-wise variation of L–P transformation matrix is computed and compared with numerical solution. Conclusion: The proposed approach is validated and proven to be applicable to both commutative and non-commutative linear quasi-periodic systems satisfying diophantine condition. Moreover, the closed form analytical expression for the L–P transformation matrix for parametrically excited linear quasi-periodic system can be obtained with this approach.