E-Book, Englisch, 330 Seiten
Romanovski / Shafer The Center and Cyclicity Problems
1. Auflage 2009
ISBN: 978-0-8176-4727-8
Verlag: Birkhäuser Boston
Format: PDF
Kopierschutz: 1 - PDF Watermark
A Computational Algebra Approach
E-Book, Englisch, 330 Seiten
ISBN: 978-0-8176-4727-8
Verlag: Birkhäuser Boston
Format: PDF
Kopierschutz: 1 - PDF Watermark
Using a computational algebra approach, this comprehensive text addresses the center and cyclicity problems as behaviors of dynamical systems and families of polynomial systems. The book gives the main properties of ideals in polynomial rings and their affine varieties followed by a discussion on the theory of normal forms and stability of differential equations. It contains numerous examples, pseudocode displays of all the computational algorithms, historical notes, nearly two hundred exercises, and an extensive bibliography, making it a suitable graduate textbook as well as research reference.
Autoren/Hrsg.
Weitere Infos & Material
1;Preface;6
2;Contents;10
3;List of Tables;12
4;Notation and Conventions;13
5;Polynomial Ideals and Their Varieties;14
5.1;1.1 Fundamental Concepts;14
5.2;1.2 The Ideal Membership Problem and Gröbner Bases;20
5.3;1.3 Basic Properties and Algorithms;37
5.4;1.4 Decomposition of Varieties;51
5.5;1.5 Notes and Complements;63
5.6;Exercises;64
6;Stability and Normal Forms;69
6.1;2.1 Lyapunov's Second Method;69
6.2;2.2 Real Normal Forms;74
6.3;2.3 Analytic and Formal Normal Forms;80
6.4;2.4 Notes and Complements;95
6.5;Exercises;96
7;The Center Problem;100
7.1;3.1 The Poincare First Return Map and the Lyapunov Numbers;102
7.2;3.2 Complexification of Real Systems, Normal Forms, and the Center Problem;107
7.3;3.3 The Center Variety;119
7.4;3.4 Focus Quantities and Their Properties;129
7.5;3.5 Hamiltonian and Reversible Systems;139
7.6;3.6 Darboux Integrals and Integrating Factors;147
7.7;3.7 Applications: Quadratic Systems and a Family of Cubic Systems;158
7.8;3.8 The Center Problem for Lienard Systems;169
7.9;3.9 Notes and Complements;175
7.10;Exercises;178
8;The Isochronicity and Linearizability Problems;186
8.1;4.1 The Period Function;186
8.2;4.2 Isochronicity Through Normal Forms and Linearizability;188
8.3;4.3 The Linearizability Quantities;202
8.4;4.4 Darboux Linearization;210
8.5;4.5 Linearizable Quadratic Centers;216
8.6;4.6 Notes and Complements;219
8.7;Exercises;220
9;Invariants of the Rotation Group;224
9.1;5.1 Properties of Invariants;225
9.2;5.2 The Symmetry Ideal and the Set of Time-Reversible Systems;240
9.3;5.3 Axes of Symmetry of a Plane System;248
9.4;5.4 Notes and Complements;255
9.5;Exercises;256
10;Bifurcations of Limit Cycles and Critical Periods;259
10.1;6.1 Bautin's Method for Bifurcation Problems;260
10.2;6.2 The Cyclicity Problem;267
10.3;6.3 The Cyclicity of Quadratic Systems and a Family of Cubic Systems;279
10.4;6.4 Bifurcations of Critical Periods;297
10.5;6.5 Notes and Complements;309
10.6;Exercises;311
11;Appendix;316
12;References;321
13;Index of Notation;330
14;Index;333
