Integrability of dynamical systems: algebra and analysis
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. This allows essential advances and generalizations of the classical results on the subject, which in many cases have been proven by the author and his collaborators. This is not, however, a mere collection of technical results based on the author's published papers. Each topic is carefully introduced and explained from basic concepts while quickly progressing to quite advanced topics. This defines the main readership of the book, which 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. The prerequisites are quite modest -- standard undergraduate courses on differential equations, differential geometry, measure theory and algebraic curves should suffice to read the book. When needed the author introduces the new concepts right in the text, together with numerous references, although it would have been more convenient to relegate some of these introductory notions to appendices. Almost each theorem is proven in generous details, and numerous remarks clarifying, explaining and supplementing the statements of the theorem are given, together with many mostly elementary (but very useful) examples for the reader to be able to apply the statements from general theorems in practice. Some of the presented results are very recent, proved only in 2016, and very often the author also poses open problems, which can serve as concrete project for graduate students. Even though there are no exercises in the book, a number of routine proofs of some theorems are left as ones. In the first chapter, the author introduces the main concepts of dynamical systems and discusses the main theorems on first integrals near a regular point. For two-dimensional manifolds the author presents a detailed discussion of the Markus-Neumann theorem. The next three chapters are the essence of the book. First, the notions of the Jacobian multiplier and the corresponding inverse are presented, and their relation to the first integrals near a regular point are explained. After giving a thorough classification of singular points, a number of results for planar differential systems and some generalization to higher-order systems are given, also, the relation of inverse Jacobian multipliers with the centre-focus problem is stated. The next key topic is the notion of Darboux integrability, and several related concepts such as Darboux polynomials and associated exponential factors, with several theorems establishing connections between them and their relation to Jacobian multipliers. Besides several original theorems due to the author and collaborators, a more elementary proof of the Darboux-Jouanolou integrability theorem on the existence of Darboux first integrals is provided, together with several generalizations of it, taking into account singularities and algebraic multiplicities. There is also a brief discussion of non-autonomous systems, with relatively straightforward generalisations of the previously discussed theorems. To introduce the notion of Liouville integrability the author carefully defines all necessary algebraic notions related to differential field extensions, although the provided bullet list may be too brief for someone who encounters such concepts for the first time. On the other hand, everything is clearly defined, and it is enough to understand the subsequent presentation of the fundamental Prelle-Singer theorem on the logarithmic form of the Liouville first integral. There are several consequences, which are discussed and proved, with particular emphasis on planar differential systems, including the relation between Darboux and Liouville integrability. The author provides an overview of some of the methods of algebraic geometry towards the problem of existence and finding the Darboux polynomials, which is a non-trivial task. Starting from the invariant algebraic curves the author defines many relevant concepts and techniques. This requires the introduction of projective spaces, nodal and nondicritical singularities, and the resolution of singularities via Puiseux series, etc. Here one should consult some standard text on algebraic curves. In the next two chapters, there are several concrete examples of differential equations borrowed from physics and other areas with a detailed analysis and lengthy discussions. In particular, Darboux polynomials for the Lorentz system are studied in great details. The same is true for the Kirchoff equations, which describe the dynamics of a rigid body in a perfect incompressible fluid, and which serves as an example of a system with algebraic and rational first integrals. Amongst other topics, the Morales-Ramis theory for homogeneous potentials of non-zero order and meromorphic first integrals is formulated, and the necessary conditions for Liouville integrability are given by restricting the eigenvalues of the Hessian matrix of the potential. A detailed description for some particular orders of the homogenous potential is provided, including the negative value. There is even a list of several concrete algorithms for computing rational first integrals. The key topics in these chapters are Poincaré's center-focus problem, and Hilbert's 16th problem for the bound of limit cycles. First, the results on the center-focus problem are stated for homogeneous quadratic polynomials systems, where the exact results are summarized in the Bautin-Kapteyn theorem, and some results on cubic homogeneous polynomials are also thoroughly discussed. Next, the relation of the center-focus problem to Darboux first integrals is given. Once more the author uses some notions of algebraic curves to derive results related to limit cycles. It is shown for example the absence of cubic limit cycles for holomorphic quadratic foliations of \(\mathbb CP^2\), and the latest results on classification of quadratic differential systems with quartic limit cycles are also given. The discussion culminates in posing and discussing Hilbert's 16h problem, and in particular its second part concerning the bound of the number of limit cycles for a planar polynomial differential system of the order \(n\). Here the author presents the solution to the weaker Hilbert's 16th problem, concerning algebraic limit cycles. Here several results closely follow and use the previous results on nodal and noncritical singularities discussed in earlier chapters. There are many interesting concrete examples, for instance the example of a differential system describing the interaction between healthy and infected cells. Finally, the last chapter brings the reader up to speed to the latest research. To study the existence of first integrals around the singularity points, the chapter starts with Poincaré's theory of normal forms, where the main tools are the Poincaré-Dulac normal forms theorem, and Poincaré's non-integrability theorem on the non-existence of first integrals. The latter theorem is extended in a few ways, by including the case of a zero eigenvalue. Several results on the number of functionally independent first integrals, and their analyticity, as well as an extension of the Poincaré-Dulac theorem for quasiperiodic systems are given. It is worth noting here that the analytic and meromorphic cases are treated differently, the latter requiring different techniques. The rest of the chapter and the final part of the book is devoted to the local theory of integrability. There are a number of extensions of classical results and technical theorems proven mainly by the author in recent years.
scientific article; zbMATH DE number 6696156
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