Integrability of Dynamical Systems: Algebra and Analysis

verfasst von: Xiang Zhang

Verlag: Springer Singapore

Buchreihe : Developments in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This is the first book to systematically state the fundamental theory of integrability and its development of ordinary differential equations with emphasis on the Darboux theory of integrability and local integrability together with their applications. It summarizes the classical results of Darboux integrability and its modern development together with their related Darboux polynomials and their applications in the reduction of Liouville and elementary integrabilty and in the center—focus problem, the weakened Hilbert 16th problem on algebraic limit cycles and the global dynamical analysis of some realistic models in fields such as physics, mechanics and biology.

Although it can be used as a textbook for graduate students in dynamical systems, it is intended as supplementary reading for graduate students from mathematics, physics, mechanics and engineering in courses related to the qualitative theory, bifurcation theory and the theory of integrability of dynamical systems.

Inhaltsverzeichnis

Frontmatter

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

Abstract

This chapter contains some definitions and properties on the first integrals of ordinary differential equations. We will prove the local integrability of differential systems around a regular point, and the global existence and regularity of the first integrals of planar smooth differential systems in the canonical regions. As a preliminary application of the integrability theory, we use it to solve linear and quasilinear partial differential equations. Lastly, we introduce a few fundamental results on Lax pairs for finding the first integrals of ordinary differential equations.

Xiang Zhang

Chapter 2. Jacobian and Inverse Jacobian Multipliers

Abstract

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 multiplicity of inverse Jacobian multipliers to study the multiplicity of a limit cycle, or of a homoclinic loop, and the cyclicity of a singularity.

Xiang Zhang

Chapter 3. Darboux and Liouvillian Integrability

Abstract

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.

Xiang Zhang

Chapter 4. Existence and Degree of Darboux Polynomials

Abstract

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.

Xiang Zhang

Chapter 5. Algebraic, Analytic and Meromorphic Integrability

Abstract

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 an algorithm to compute the rational first integrals and the Darboux polynomials of polynomial differential systems.

Xiang Zhang

Chapter 6. Applications of the Darboux Theory of Integrability

Abstract

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 this theory to specific models in two and higher dimensions.

Xiang Zhang

Chapter 7. Local Integrability of Differential Systems

Abstract

In Chap. 1 we proved that for \(k\in \mathbb N\cup \{\infty ,\omega \}\), an n-dimensional \(C^k\) autonomous differential system always has \(n-1\) functionally independent \(C^k\) first integrals in a neighborhood of a regular point, where the first integrals are independent of the independent variable of the system. This chapter will concentrate on the existence of analytic or formal first integrals of analytic differential systems in a neighborhood of a singularity, with an emphasis on the varieties and the existence of analytic normalizations of analytic integrable (or partially integrable) differential systems. We will also introduce the local theory of Darboux integrability of local analytic or formal differential systems.

Xiang Zhang

Backmatter

Titel: Integrability of Dynamical Systems: Algebra and Analysis
verfasst von: Xiang Zhang
Verlag: Springer Singapore
Electronic ISBN: 978-981-10-4226-3
Print ISBN: 978-981-10-4225-6
DOI: https://doi.org/10.1007/978-981-10-4226-3