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.

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

2021 ◽  
pp. 107754632199356 ◽  
Author(s):  
Susheelkumar C Subramanian ◽  
Peter MB Waswa ◽  
Sangram Redkar
Keyword(s):  
Closed Form ◽  
Periodic System ◽  
Linear Time ◽  
Closed Form Expression ◽  
Periodic Systems ◽  
Form Expression ◽  
State Augmentation ◽  
Time Invariant ◽  
Time Periodic ◽  
Time Invariant System

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.

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.

Lyapunov Perron Transformation for Linear Quasi-Periodic Systems

2020 ◽  
Author(s):  
Susheelkumar C. Subramanian ◽  
Sangram Redkar ◽  
Peter Waswa
Keyword(s):  
Periodic System ◽  
Floquet Theory ◽  
Normal Forms ◽  
Closed Form Expression ◽  
Periodic Systems ◽  
Integration Technique ◽  
Form Expression ◽  
P Transformation ◽  
State Augmentation ◽  
Time Invariant

Abstract It is known that a Lyapunov Perron (L-P) transformation converts a quasi-periodic system into a reduced system with a time-invariant coefficient. Though a closed form expression for L-P transformation matrix is missing in the literature, the application of combination of multiple theories would aid in such transformation. In this work, the authors have worked on extending the Floquet theory to find L-P transformation. As an example, a commutative system with linear quasi-periodic coefficients is transformed into a system with time-invariant coefficient analytically. Furthermore, for non-commutative systems, similar results are obtained in this work, with the help of an intuitive state augmentation and Normal Forms technique. The results of the reduced system are compared with the numerical integration technique for validation.

A Simple Approach for the Computation of Lyapunov-Floquet Transformations for the Mathieu Equation

2021 ◽  
Author(s):  
Ashu Sharma
Keyword(s):  
Periodic System ◽  
Linear Time ◽  
Mathieu Equation ◽  
Adjoint System ◽  
Simple Approach ◽  
Periodic Systems ◽  
Linear Ordinary Differential Equations ◽  
Constant Coefficients ◽  
Time Invariant ◽  
Time Periodic

Abstract Lyapunov-Floquet (L-F) transformations reduce linear ordinary differential equations with time-periodic coefficients (so-called linear time-periodic systems) to equations with constant coefficients. The present work proposes a simple approach to construct L-F transformations. The solution of a linear time-periodic system can be expressed as a product of an exponential term and a periodic term. Using this Floquet form of a solution, the ordinary differential equation corresponding to a linear time-periodic system reduces to an eigenvalue problem. Next, eigenanalysis is performed to obtain the general solution and subsequently, the state transition matrix of the time-periodic system is constructed. Then, the Lyapunov-Floquet theorem is used to compute L-F transformation. The inverse of L-F transformation is determined by defining the adjoint system to the time-periodic system. Mathieu equation is investigated in this work and L-F transformations and their inverse are generated for stable and unstable cases. These transformations are very useful in the design of controllers using time-invariant methods and in the bifurcation studies of nonlinear time-periodic systems.

scholarly journals Terminal Wiener Index of Fibonacci trees

2020 ◽  
Vol 9 (3) ◽  
pp. 2533-2535
Keyword(s):  
Closed Form ◽  
Linear Time ◽  
Wiener Index ◽  
Time Algorithm ◽  
Closed Form Expression ◽  
Linear Time Algorithm ◽  
Form Expression ◽  
Sum Of Distances

The terminal Wiener index of a tree is defined as the sum of distances between all leaf pairs of T. We derive closed form expression for the terminal Wiener index of fibonacci trees. We also describe a linear time algorithm to compute terminal Wiener index of a tree.

Reducibility and Analysis of Linear Quasi-Periodic Systems Via Normal Forms

2020 ◽  
Vol 15 (9) ◽  
Author(s):  
Peter M. B. Waswa ◽  
Sangram Redkar
Keyword(s):  
Periodic System ◽  
Constant Coefficient ◽  
Floquet Theory ◽  
Normal Forms ◽  
Time History ◽  
Mathieu Equation ◽  
Periodic Systems ◽  
State Augmentation ◽  
The Stability ◽  
Averaging Techniques

Abstract This article introduces a technique to accomplish reducibility of linear quasi-periodic systems into constant-coefficient linear systems. This is consistent with congruous proofs common in literature. Our methodology is based on Lyapunov–Floquet transformation, normal forms, and enabled by an intuitive state augmentation technique that annihilates the periodicity in a system. Unlike common approaches, the presented approach does not employ perturbation or averaging techniques and does not require a periodic system to be approximated from the quasi-periodic system. By considering the undamped and damped linear quasi-periodic Hill-Mathieu equation, we validate the accuracy of our approach by comparing the time-history behavior of the reduced linear constant-coefficient system with the numerically integrated results of the initial quasi-periodic system. The two outcomes are shown to be in exact agreement. Consequently, the approach presented here is demonstrated to be accurate and reliable. Moreover, we employ Floquet theory as part of our analysis to scrutinize the stability and bifurcation properties of the undamped and damped linear quasi-periodic system.

scholarly journals Synthesizing of Planar and Conformal Double-Sided Parallel-Cross Dipoles FSS Based on Closed-Form Expression

IEEE Access ◽  
2021 ◽  
pp. 1-1
Author(s):  
Yassine Zouaoui ◽  
Larbi Talbi ◽  
Khelifa Hettak ◽  
Naresh K. Darimireddy
Keyword(s):  
Closed Form ◽  
Closed Form Expression ◽  
Form Expression

scholarly journals Linear Time-Periodic System Identification with Grouped Atomic Norm Regularization

2020 ◽  
Vol 53 (2) ◽  
pp. 1237-1242
Author(s):  
Mingzhou Yin ◽  
Andrea Iannelli ◽  
Mohammad Khosravi ◽  
Anilkumar Parsi ◽  
Roy S. Smith
Keyword(s):  
System Identification ◽  
Periodic System ◽  
Linear Time ◽  
Atomic Norm ◽  
Time Periodic

Distribution of Consensus in a Broadcasting-based Consensus-forming Algorithm

2021 ◽  
Vol 48 (3) ◽  
pp. 91-96
Author(s):  
Shigeo Shioda
Keyword(s):  
Distribution Function ◽  
Probability Density Function ◽  
Closed Form ◽  
Probability Density ◽  
Infinite Number ◽  
Stable Distribution ◽  
Random Variable ◽  
Cauchy Distribution ◽  
Closed Form Expression ◽  
Form Expression

The consensus achieved in the consensus-forming algorithm is not generally a constant but rather a random variable, even if the initial opinions are the same. In the present paper, we investigate the statistical properties of the consensus in a broadcasting-based consensus-forming algorithm. We focus on two extreme cases: consensus forming by two agents and consensus forming by an infinite number of agents. In the two-agent case, we derive several properties of the distribution function of the consensus. In the infinite-numberof- agents case, we show that if the initial opinions follow a stable distribution, then the consensus also follows a stable distribution. In addition, we derive a closed-form expression of the probability density function of the consensus when the initial opinions follow a Gaussian distribution, a Cauchy distribution, or a L´evy distribution.

scholarly journals Colored HOMFLY-PT for hybrid weaving knot $$ {\hat{\mathrm{W}}}_3 $$(m, n)

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Vivek Kumar Singh ◽  
Rama Mishra ◽  
P. Ramadevi
Keyword(s):  
Closed Form ◽  
Topological Strings ◽  
Chebyshev Polynomials ◽  
Fibonacci Numbers ◽  
Infinite Family ◽  
Closed Form Expression ◽  
Two Dimensional ◽  
Laurent Polynomials ◽  
Form Expression ◽  
Torus Knots

Abstract Weaving knots W(p, n) of type (p, n) denote an infinite family of hyperbolic knots which have not been addressed by the knot theorists as yet. Unlike the well known (p, n) torus knots, we do not have a closed-form expression for HOMFLY-PT and the colored HOMFLY-PT for W(p, n). In this paper, we confine to a hybrid generalization of W(3, n) which we denote as $$ {\hat{W}}_3 $$ W ̂ 3 (m, n) and obtain closed form expression for HOMFLY-PT using the Reshitikhin and Turaev method involving $$ \mathrm{\mathcal{R}} $$ ℛ -matrices. Further, we also compute [r]-colored HOMFLY-PT for W(3, n). Surprisingly, we observe that trace of the product of two dimensional $$ \hat{\mathrm{\mathcal{R}}} $$ ℛ ̂ -matrices can be written in terms of infinite family of Laurent polynomials $$ {\mathcal{V}}_{n,t}\left[q\right] $$ V n , t q whose absolute coefficients has interesting relation to the Fibonacci numbers $$ {\mathrm{\mathcal{F}}}_n $$ ℱ n . We also computed reformulated invariants and the BPS integers in the context of topological strings. From our analysis, we propose that certain refined BPS integers for weaving knot W(3, n) can be explicitly derived from the coefficients of Chebyshev polynomials of first kind.

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