Integrability of Dynamical Systems: Algebra and Analysis

1. Front Matter

   Pages i-xv

   PDF

2. The Fundamentals of the Theory of Integrability of Differential Systems

   • Xiang Zhang

   Pages 1-33

3. Jacobian and Inverse Jacobian Multipliers

   • Xiang Zhang

   Pages 35-88

4. Darboux and Liouvillian Integrability

   • Xiang Zhang

   Pages 89-148

5. Existence and Degree of Darboux Polynomials

   • Xiang Zhang

   Pages 149-195

6. Algebraic, Analytic and Meromorphic Integrability

   • Xiang Zhang

   Pages 197-252

7. Applications of the Darboux Theory of Integrability

   • Xiang Zhang

   Pages 253-285

8. Local Integrability of Differential Systems

   • Xiang Zhang

   Pages 287-351

9. Back Matter

   Pages 353-380

   PDF

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.

• integrability
• Jacobian multipliers
• Darboux integrability
• Liouvillian integrability
• Darboux polynomials
• Local integrability
• ordinary differential equations

“This excellent book is focused on the algebraic and analytic aspects of the integrability of ordinary differential equations and of dynamical systems. … The book is a very good presentation of recent results on the subject that provides an important service to the mathematical community and should be a useful reference for people who work or will work in the future in this beautiful area.” (Jaume Giné, Mathematical Reviews, January, 2018)

“The book is devoted to the theory of dynamical and integrable systems, focusing primarily on recent developments and modern techniques, relying especially on the theory of algebraic curves. … will be mainly useful for advanced undergraduate and beginning graduate students interested in the modern theory of dynamical and integrable systems, and who want to quickly acquire the necessary knowledge in basics and be brought up to speed with the latest research.” (Arsen Melikyan, zbMATH 1373.37053, 2017)

• School of Mathematical Science, Shanghai Jiao Tong University , Shanghai, China

  Xiang Zhang

Xiang Zhang is a distinguished professor of Shanghai Jiao Tong University, China. He obtained his PhD degree in 1997 from Nanjing University, China.His main interest is in the field of qualitative theory, bifurcation theory and the theory of integrability of Ordinary Differential Equations and Dynamical Systems.

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