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Integrability of Dynamical Systems: Algebra and Analysis

Part of the Developments in Mathematics Book Series (volume 47)

DOI: 10.1007/978-981-10-4226-3

ISBN: 978-981-10-4225-6

Publisher: Springer Singapore - Springer Journals

Mar 31, 2017

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Chapter 1The Fundamentals of the Theory of Integrability of Differential Systems

Zhang, Xiang

Pages: 1 –33

[This chapter contains some definitions and properties on the first integrals of ordinary differential equations. We will prove the local integrability of differential systemsDifferential system around a regular point, and the global existence and regularity of the first integrals of planar...
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Chapter 2Jacobian and Inverse Jacobian Multipliers

Zhang, Xiang

Pages: 35 –88

[This chapter is mainly concerned with the existence and regularity of the Jacobian and the inverse Jacobian multipliers of differential systems near a singularity, a periodic orbit, or a polycycle. We will also use the vanishing multiplicityVanishing multiplicityof inverse Jacobian multiplier of...
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Chapter 3Darboux and Liouvillian Integrability

Zhang, Xiang

Pages: 89 –148

[Darboux and Liouvillian integrability is mainly concerned with algebraic aspects of the integrability of differential systems, which is related to many subjects, such as real and complex analysis, algebraic geometry, differential algebra, differential Galois theory, and so on.]
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Chapter 4Existence and Degree of Darboux Polynomials

Zhang, Xiang

Pages: 149 –195

[This chapter presents results on the degree and existence of Darboux polynomials with an emphasis on invariant algebraic curves. We also introduce some tools and methods for characterizing the Darboux polynomials of polynomial vector fields.]
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Chapter 5Algebraic, Analytic and Meromorphic Integrability

Zhang, Xiang

Pages: 197 –252

[This chapter studies the polynomial and analytic integrability of some concrete physical models, the polynomial and rational integrability of Hamiltonian systems, and the meromorphic integrability of differential systems near a given orbit via the differential Galois group. Finally, we present...
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Chapter 6Applications of the Darboux Theory of Integrability

Zhang, Xiang

Pages: 253 –285

[The Darboux theory of integrability has been successively applied to study the dynamics of polynomial differential systems. This chapter will focus on the center-focus and the limit cycle problems of planar polynomial differential systems. In the last section we will also present applications of...
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Chapter 7Local Integrability of Differential Systems

Zhang, Xiang

Pages: 287 –351

[In Chap. 1 we proved that for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt}...

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