Computing Poincaré-Lyapunov Constants via Carleman Linearization

2019 ◽  
Author(s):  
Csanád Árpád Hubay ◽  
Tamás Kalmár-Nagy
Keyword(s):  
Hopf Bifurcation ◽  
Linear Algebra ◽  
Fredholm Alternative ◽  
Solvability Conditions ◽  
Carleman Matrix ◽  
Algebraic Properties ◽  
Lyapunov Constants

Abstract Using Carleman linearization an approximation is given for the solution of a system at Hopf bifurcation. The values of the Poincaré-Lyapunov constants (whether they are zero or not) affect the linear algebraic properties of the Carleman matrix and they appear in solvability conditions (through the Fredholm alternative). We provide a linear algebra based algorithm to compute the Poincaré-Lyapunov constants.

Hopf Bifurcation in Gas Journal Bearings

1993 ◽  
Vol 46 (7) ◽  
pp. 392-398 ◽  
Author(s):  
K. Czołczyn´ski
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Bifurcation Theory ◽  
Journal Bearing ◽  
Equations Of Motion ◽  
Journal Bearings ◽  
Fredholm Alternative ◽  
Stiffness Coefficients ◽  
Nonlinear Equations Of Motion ◽  
Gas Journal Bearing

This paper reviews a numerical investigation of the problem of small self-excited vibrations in gas journal bearings. The method of analysis is based on the Hopf bifurcation theory, in which the approximate periodic solutions of nonlinear equations of motion are computed using the Fredholm alternative. This theory enables us to construct the bifurcating periodic solutions and to determine their stability. The equations of motion of the investigated gas journal bearing have been formulated after estimating the damping and stiffness coefficients of a gas film. For this purpose, a new method of identification has been proposed.

CENTER PROBLEM AND MULTIPLE HOPF BIFURCATION FOR THE Z5-EQUIVARIANT PLANAR POLYNOMIAL VECTOR FIELDS OF DEGREE 5

2009 ◽  
Vol 19 (06) ◽  
pp. 2115-2121 ◽  
Author(s):  
YIRONG LIU ◽  
JIBIN LI
Keyword(s):  
Vector Field ◽  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Vector Fields ◽  
Center Problem ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Perturbed System ◽  
Lyapunov Constants

This paper proves that a Z5-equivariant planar polynomial vector field of degree 5 has at least five symmetric centers, if and only if it is a Hamltonian vector field. The characterization of a center problem is completely solved. The shortened expressions of the first four Lyapunov constants are given. Under small Z5-equivariant perturbations, the conclusion that the perturbed system has at least 25 limit cycles with the scheme 〈5 ∐ 5 ∐ 5 ∐ 5 ∐ 5〉 is rigorously proved.

scholarly journals Dual Numbers Approach in Multiaxis Machines Error Modeling

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Jaroslav Hrdina ◽  
Petr Vašík
Keyword(s):  
Differential Geometry ◽  
Linear Algebra ◽  
Error Modeling ◽  
Geometric Errors ◽  
Dual Numbers ◽  
Algebraic Properties ◽  
Classes Of Matrices ◽  
Multiaxis Machines ◽  
Special Classes

Multiaxis machines error modeling is set in the context of modern differential geometry and linear algebra. We apply special classes of matrices over dual numbers and propose a generalization of such concept by means of general Weil algebras. We show that the classification of the geometric errors follows directly from the algebraic properties of the matrices over dual numbers and thus the calculus over the dual numbers is the proper tool for the methodology of multiaxis machines error modeling.

On Maps Preserving Products

2005 ◽  
Vol 48 (3) ◽  
pp. 355-369 ◽  
Author(s):  
M. A. Chebotar ◽  
W.-F. Ke ◽  
P.-H. Lee ◽  
L.-S. Shiao
Keyword(s):  
Functional Analysis ◽  
Linear Algebra ◽  
Point Of View ◽  
Algebraic Properties ◽  
Theoretic Point

AbstractMaps preserving certain algebraic properties of elements are often studied in Functional Analysis and Linear Algebra. The goal of this paper is to discuss the relationships among these problems from the ring-theoretic point of view.

Matrix Completions, Moments, and Sums of Hermitian Squares

2011 ◽  
Author(s):  
Mihály Bakonyi ◽  
Hugo J. Woerdeman
Keyword(s):  
Linear Algebra ◽  
Undergraduate Students ◽  
Complex Analysis ◽  
Measure Theory ◽  
Complex Function ◽  
Complex Function Theory ◽  
Extensive Discussion ◽  
The Subject ◽  
Processing Control ◽  
Developing Area

Intensive research in matrix completions, moments, and sums of Hermitian squares has yielded a multitude of results in recent decades. This book provides a comprehensive account of this quickly developing area of mathematics and applications and gives complete proofs of many recently solved problems. With MATLAB codes and more than two hundred exercises, the book is ideal for a special topics course for graduate or advanced undergraduate students in mathematics or engineering, and will also be a valuable resource for researchers. Often driven by questions from signal processing, control theory, and quantum information, the subject of this book has inspired mathematicians from many subdisciplines, including linear algebra, operator theory, measure theory, and complex function theory. In turn, the applications are being pursued by researchers in areas such as electrical engineering, computer science, and physics. The book is self-contained, has many examples, and for the most part requires only a basic background in undergraduate mathematics, primarily linear algebra and some complex analysis. The book also includes an extensive discussion of the literature, with close to six hundred references from books and journals from a wide variety of disciplines.

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