An Approximate Analysis of Quasi-Periodic Systems via Floquét Theory

2017 ◽  
Author(s):  
Ashu Sharma ◽  
Subhash C. Sinha
Keyword(s):  
Parametric Excitation ◽  
Periodic System ◽  
Floquet Theory ◽  
Periodic Systems ◽  
Frequency Spectra ◽  
Periodic Coefficients ◽  
Approximate Approach ◽  
Varying Coefficients ◽  
P Transformation ◽  
The Stability

Parametrically excited systems are generally represented by a set of linear/nonlinear ordinary differential equations with time varying coefficients. In most cases, the linear systems have been modeled by Mathieu or Hill equations (periodic coefficients) because their stability and response can be determined by Floquét theory. However, in many cases the parametric excitation is not periodic but consists of frequencies that are incommensurate, making them quasi-periodic. Unfortunately, there is no complete theory for linear dynamic systems with quasi-periodic coefficients. Motivated by this fact, in this work, an approximate approach has been proposed to determine the stability and response of quasi-periodic systems. Although Floquét theory is applicable only to periodic systems, it is suggested here that a quasi-periodic system may be replaced by a periodic system with an appropriate large principal period and thus making it suitable for an application of the Floquét theory. Based on this premise, a systematic approach has been developed and applied to two typical quasi-periodic systems. The approximate boundaries in stability charts obtained from the proposed method are extremely close to the exact boundaries of the original quasi-periodic equations. The exact boundaries are detected by computing the maximal Lyapunov exponents. Further, the frequency spectra of solutions generated near approximate and exact boundaries are found to be almost identical ensuring a high degree of accuracy. The coefficients of the parametric excitation terms are not necessarily small in all cases. ‘Instability loops’ or ‘Instability pockets’ that appear in the stability diagram of Meissner’s equation are also observed in one case presented here. The proposed approximate approach would allow one to construct Lyapunov-Perron (L-P) transformation matrices that reduce quasi-periodic systems to systems whose linear parts are time-invariant. The L-P transformation would pave the way for controller design and bifurcation analysis of quasi-periodic systems.

An Approximate Analysis of Quasi-Periodic Systems Via Floquét Theory

2017 ◽  
Vol 13 (2) ◽  
Author(s):  
Ashu Sharma ◽  
S. C. Sinha
Keyword(s):  
Parametric Excitation ◽  
Periodic System ◽  
Floquet Theory ◽  
Picard Iteration ◽  
Periodic Systems ◽  
Transition Matrices ◽  
Frequency Spectra ◽  
Periodic Coefficients ◽  
Stability Diagrams ◽  
The Stability

Parametrically excited linear systems with oscillatory coefficients have been generally modeled by Mathieu or Hill equations (periodic coefficients) because their stability and response can be determined by Floquét theory. However, in many cases, the parametric excitation is not periodic but consists of frequencies that are incommensurate, making them quasi-periodic. Unfortunately, there is no complete theory for linear dynamic systems with quasi-periodic coefficients. Motivated by this fact, in this work, an approximate approach has been proposed to determine the stability and response of quasi-periodic systems. It is suggested here that a quasi-periodic system may be replaced by a periodic system with an appropriate large principal period and thus making it suitable for an application of the Floquét theory. Based on this premise, a systematic approach has been developed and applied to three typical quasi-periodic systems. The approximate boundaries in stability charts obtained from the proposed method are very close to the exact boundaries of original quasi-periodic equations computed numerically using maximal Lyapunov exponents. Further, the frequency spectra of solutions generated near approximate and exact boundaries are found to be almost identical ensuring a high degree of accuracy. In addition, state transition matrices (STMs) are also computed symbolically in terms of system parameters using Chebyshev polynomials and Picard iteration method. Stability diagrams based on this approach are found to be in excellent agreement with those obtained from numerical methods. The coefficients of parametric excitation terms are not necessarily small in all cases.

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.

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.

Control of Nonlinear Time-Periodic Systems Via Feedback Linearization

2005 ◽  
Author(s):  
Yandong Zhang ◽  
S. C. Sinha
Keyword(s):  
Control System ◽  
Periodic System ◽  
Feedback Linearization ◽  
Linear Method ◽  
Sufficient Conditions ◽  
Periodic Systems ◽  
Computation Method ◽  
Normal Form Theory ◽  
Periodic Coefficients ◽  
Time Periodic

The problem of designing controllers for nonlinear time periodic systems is addressed. The idea is to find proper coordinate transformations and state feedback under which the original system can be (approximately) transformed into a linear control system. Then a controller can be designed using the well-known linear method to guarantee the stability of the system. We propose two approaches for the feedback linearization of the nonlinear time periodic system. The first approach is designed to achieve local control of nonlinear systems with periodic coefficients desired to be driven either to a periodic orbit or to a fixed point. In this case the system equations can be represented by a quasi-linear system containing nonlinear monomials with periodic coefficients. Using near identity transformations and normal form theory, the original close loop problem is approximately transformed into a linear time periodic system with unknown gains. Then by using a symbolic computation method, the Floquet multipliers are placed in the desired locations in order to determine the control gains. We also give the sufficient conditions under which the system is feedback linearizable up to the rth order. The second approach is a generalization of the classical exact feedback linearization method for autonomous systems but applicable to general time-periodic affine systems. By defining a time-dependent Lie operator, the input-output nonlinear time periodic problem is transformed into a linear autonomous problem for which control system can be designed easily. A sufficient condition under which the system is feedback linearizable is also given.

A Preliminary Investigation of the Dynamic Stability of Flexible Manipulators Performing Repetitive Tasks

1986 ◽  
Vol 108 (3) ◽  
pp. 206-214 ◽  
Author(s):  
D. A. Streit ◽  
C. M. Krousgrill ◽  
A. K. Bajaj
Keyword(s):  
Parametric Excitation ◽  
Floquet Theory ◽  
Equations Of Motion ◽  
Preliminary Investigation ◽  
Zero Solution ◽  
Flexible Manipulator ◽  
Governing Equations ◽  
Repetitive Tasks ◽  
The Stability ◽  
Two Degree Of Freedom

The governing equations of motion for the compliant coordinates describing a flexible manipulator performing repetitive tasks contain parametric excitation terms. The stability of the zero solution to these equations is investigated using Floquet theory. Analytical and numerical results are presented for a two-degree-of-freedom model of a manipulator with one prismatic joint and one revolute joint.

Lyapunov Perron Transformation for Quasi-Periodic Systems and its Applications

2021 ◽  
pp. 1-26
Author(s):  
Susheelkumar Cherangara Subramanian ◽  
Sangram Redkar
Keyword(s):  
Periodic System ◽  
Normal Forms ◽  
Identity Transformation ◽  
Periodic Systems ◽  
Invariant System ◽  
External Excitation ◽  
P Transformation ◽  
State Augmentation ◽  
Time Invariant ◽  
Time Invariant System

Abstract 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.

A Study of the Parametrically Excited Behavior of Passenger and Freight Railway Vehicles Using Linear Models

1991 ◽  
Vol 113 (2) ◽  
pp. 336-338 ◽  
Author(s):  
J. Lieh ◽  
I. Haque
Keyword(s):  
Parametric Resonance ◽  
Parametric Excitation ◽  
Floquet Theory ◽  
Linear Models ◽  
Stability Boundaries ◽  
Parametrically Excited ◽  
Critical Speeds ◽  
Railway Vehicles ◽  
Principal Parametric Resonance ◽  
The Stability

This paper presents a study of the parametrically excited behavior of passenger and freight vehicles on tangent track due to harmonic variations in conicity using linear models. The effect of primary and secondary stiffnesses on parametric excitation is also studied. Floquet theory is used to find the stability boundaries. The results show that wavelengths associated with conicity variation that are in the vicinity of half the kinematic wavelengths of the vehicles can lead to significant reductions in critical speeds. Results also show that the primary and warp stiffnesses can affect the severity of principal parametric resonance depending on the vehicle models and magnitude of stiffnesses chosen.

Impulsive Parametric Excitation: Theory

1972 ◽  
Vol 39 (2) ◽  
pp. 551-558 ◽  
Author(s):  
C. S. Hsu
Keyword(s):  
Stability Analysis ◽  
General Theory ◽  
Parametric Excitation ◽  
Periodic Systems ◽  
Stability Criteria ◽  
Dirac Delta ◽  
Special Cases ◽  
Delta Functions ◽  
Analytic Stability ◽  
The Stability

Given in this paper is the development of a theory for dynamical systems subjected to periodic impulsive parametric excitations. By periodic impulsive parametric excitation we mean those excitations representable by periodic coefficients which consist of sequences of Dirac delta functions. It turns out that for this class of periodic systems the stability analysis can be carried out in a remarkably simple and general manner without approximation. In the paper, after giving the general theory, many special cases are examined. In many instances simple and closed-form analytic stability criteria can be easily established.

scholarly journals Nonlinear parametric excitation of an evolutionary dynamical system

2011 ◽  
Vol 226 (8) ◽  
pp. 1912-1920 ◽  
Author(s):  
Rocio E Ruelas ◽  
David G Rand ◽  
Richard H Rand
Keyword(s):  
Game Theory ◽  
Differential Equations ◽  
Parametric Excitation ◽  
Evolutionary Dynamics ◽  
Evolutionary Game ◽  
Periodic Coefficients ◽  
The Stability ◽  
Social Applications ◽  
Classical Game ◽  
Classical Game Theory

Nonlinear parametric excitation refers to the nonlinear analysis of a system of ordinary differential equations with periodic coefficients. In contrast to linear parametric excitation, which offers determinations of the stability of equilibria, nonlinear parametric excitation has as its goal the structure of the phase space, as given by a portrait of the Poincare map. In this article, perturbation methods and numerical integration are applied to the replicator equation with periodic coefficients, being a model from evolutionary game theory where evolutionary dynamics are added to classical game theory using differential equations. In particular, we study evolution in the Rock–Paper–Scissors game, which has biological and social applications. Here, periodic coefficients could represent seasonal variation.

In-Plane Parametric Instability of a Rigid Body With a Dual-Rotor System

2015 ◽  
Author(s):  
Allen Anilkumar ◽  
V. Kartik
Keyword(s):  
Rigid Body ◽  
Floquet Theory ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Parametric Instability ◽  
Harmonic Balance Method ◽  
Frequency Spectra ◽  
Stability Diagrams ◽  
The Stability ◽  
Lumped Masses

Rotating machines can be modeled at a basic level using lumped masses that are rotating about and attached using springs to an axis. Even such seemingly simple system can exhibit rich dynamics in the presence of time-varying terms in the governing differential equations. This paper investigates the dynamics of a rigid body with two attached rotors that rotate in the same plane. The system is parametrically-excited and the equations of motion are periodic in both rotor frequencies. The frequency spectra of the time responses show distinct side-band structures centered about the unforced natural frequencies. In addition to classical resonances, the stability diagrams generated using Floquet theory reveal instabilities at unexpected combinations of the forcing and natural frequencies. The harmonic balance method is employed to verify the stability boundaries obtained using Floquet theory. The study reveals safe regimes of parameter combinations that can help prevent the onset of instability in such systems.

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