Unification of Poincar\xe9 and Floquet Theory for Time Periodic Systems

2021 ◽  
Author(s):  
Susheelkumar Cherangara Subramanian ◽  
Sangram Redkar
Keyword(s):  
Floquet Theory ◽  
Periodic Systems ◽  
Time Periodic

Comparison of Poincaré Normal Forms and Floquet Theory for Analysis of Linear Time Periodic Systems

2020 ◽  
Vol 16 (1) ◽  
Author(s):  
Susheelkumar C. Subramanian ◽  
Sangram Redkar
Keyword(s):  
Floquet Theory ◽  
Normal Forms ◽  
Linear Time ◽  
Mathieu Equation ◽  
Temporal Variations ◽  
Periodic Systems ◽  
Comparative Results ◽  
Time Invariant ◽  
Identity Transformations ◽  
Time Periodic

Abstract In this work, the authors draw comparisons between the Floquet theory and Normal Forms technique and apply them towards the investigation of stability bounds for linear time periodic systems. Though the Normal Forms technique has been predominantly used for the analysis of nonlinear equations, in this work, the authors utilize it to transform a linear time periodic system to a time-invariant system, similar to the Lyapunov–Floquet (L–F) transformation. The authors employ an intuitive state augmentation technique, modal transformation, and near identity transformations to facilitate the application of time-independent Normal Forms. This method provides a closed form analytical expression for the state transition matrix (STM). Additionally, stability analysis is performed on the transformed system and the comparative results of dynamical characteristics and temporal variations of a simple linear Mathieu equation are also presented in this work.

Unification of Poincaré and Floquet Theory for Time Periodic Systems

2020 ◽  
Author(s):  
Susheelkumar C. Subramanian ◽  
Sangram Redkar
Keyword(s):  
Closed Form ◽  
Periodic System ◽  
Floquet Theory ◽  
Linear Time ◽  
Mathieu Equation ◽  
Transformation Matrix ◽  
Closed Form Expression ◽  
Periodic Systems ◽  
Form Expression ◽  
Time Periodic

Abstract As per Floquet theory, a transformation matrix (Lyapunov Floquet transformation matrix) converts a linear time periodic system to a linear time-invariant one. Though a closed form expression for such a matrix was missing in the literature, this method has been widely used for studying the dynamical stability of a time periodic system. In this paper, the authors have derived a closed form expression for the Lyapunov Floquet (L-F) transformation matrix analytically using intuitive state augmentation, Modal Transformation and Normal Forms techniques. The results are tested and validated with the numerical methods on a Mathieu equation with and without damping. This approach could be applied to any linear time periodic systems.

Linear Time-Periodic Systems with Exceptional Points of Degeneracy

2019 ◽  
Author(s):  
Hamidreza Kazemi ◽  
Mohamed Y. Nada ◽  
Tarek Mealy ◽  
Ahmed F. Abdelshafy ◽  
Filippo Capolino
Keyword(s):  
Linear Time ◽  
Periodic Systems ◽  
Exceptional Points ◽  
Time Periodic
Sign in / Sign up

Export Citation Format

Share Document