Review of “Integrability of Dynamical Systems: Algebra and Analysis” by Xiang Zhang

By S.C. Coutinho
Integrability of dynamical systems: algebra and analysis
Integrability of Dynamical Systems: Algebra and Analysis
Xiang Zhang
380 pp. (2017)
ISBN:  978-981-10-4225-6
Reviewed by: S.C. Coutinho
Departamento de Ciência da Computação
Universidade Federal do Rio de Janeiro, Brazil
Email: collier (at)

At first sight, one may get the impression that the title of this book makes quite clear what it is about; unfortunately, different mathematicians understand different things when they hear that a system of differential equations is integrable. But, although the word has different shades of meaning in different chapters, one can safely say that, throughout the book, it means that the system has at least one first integral. A quick look at the index confirms both statements: it lists 20 different combinations in which the expression first integral is used. Not surprisingly, the definition and basic properties of these integrals are one of the major themes of chapter 1.

The search for first integrals is almost as old as the study of differential equations itself. Introduced by A. C. Clairaut in 1734, and further developed by L. Euler in his Institutionum Calculi Integralis of 1768, the well-known method of integrating factors was the first systematic algorithm for finding first integrals. In chapter 2 we are introduced to Jacobian multipliers, which are generalizations of integrating factors to systems of higher dimension, and their inverses. This chapter also includes applications of these concepts to the centre-focus problem and to Hopf bifurcation.

The next greatest contribution to the search for first integrals was, arguably, Gaston Darboux’s Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré, published in 1878. Influenced by recent developments in projective geometry, Darboux showed how a system of differential equations of dimension two with polynomial coefficients can be translated as an incidence correspondence. This allowed him to give a purely algebraic interpretation to what it means for a projective algebraic curve to be a solution of the differential system. To limit the search of solutions to curves defined by the vanishing of polynomials may seem like an unnecessary restriction; however, Darboux also showed that, if there are enough solutions of this type, they can be used to compute a multivalued first integral of the system of differential equations.

Before we proceed, it is convenient to introduce some terminology. Let \(F\) be a polynomial vector field in \(\mathbb C^n\). A polynomial \(P\) in \(n\) variables is a Darboux polynomial of the system \(\dot{X} = F(X)\) if its Lie derivative with respect to \(F\) is a (polynomial) multiple of \(P\). In this case we also say that the set of points of \(\mathbb C^n\) where \(P\) vanishes is an invariant hypersurface of \(\dot{X} = F(X)\). When \(F\) is defined by homogeneous polynomials of the same degree, it is natural to take \(P\) to be also homogeneous and to consider the hypersurface it defines in the complex projective space of dimension \(n-1\).

In 1891 Poincaré published a paper called Sur l’integration algébrique des équations differentielles du 1er ordre, in whose short first paragraph he mentions a problem that remains unsolved to this day:

To determine if a differential equation of the first order and the first degree is algebraically integrable, it is evidently enough to find an upper bound on the degree of the integral; then all that remains is to perform some purely algebraic computations.

In Poincaré’s terminology, an equation is algebraically integrable if it has a rational first integral; the problem consists in finding an upper bound on the degree of the polynomials that define this function. As Poincaré explains in more detail a little later in the same paper, the bound applies only to indecomposable rational functions, which are those that cannot be written as a composition of two rational functions, one of which is a first integral.

Several basic texts on analysis and differential equations published at the end of the 19th and beginning of the 20th century introduced Darboux’s method. These include C. Jordan’s Cours d’Analyse de l’Ecole Polytechnique and Ince’s Ordinary differential equations. However, by the 1930s, interest had shifted from finding explicit solutions of differential equations, to the questions of stability that we now associate with the theory of dynamical systems. As a consequence, Darboux’s work was almost completely forgotten, until it was re-worked in the language of modern algebraic geometry, and vastly generalized, by J.-P. Jouanolou in his 1979 monograph Equations de Pfaff algébriques [1]. One of the new results in Jouanolou’s monograph was the characterization of systems of polynomial differential equations with a rational first integral as those with infinitely many Darboux polynomials [1, Théoréme 3.3, p. 102]. In particular, if a system does not have a rational first integral then there must be an upper bound on the degree of its (irreducible) Darboux polynomials.

By the time Jouanolou’s monograph appeared, the first versions of Risch’s algorithm on the integration of elementary functions, published ten years earlier, were being implemented in computer algebra systems. So the time seemed ripe to tackle the symbolic integration of differential equations. The case of systems of first order equations was solved by M. J. Prelle and M. F. Singer in a 1983 paper, where they propose an algorithm to find elementary first integrals.

Four of the seven chapters of the book under review are concerned with results directly related to Darboux’s 1878 paper. The first of these is chapter 3, where Darboux polynomials are defined, and that also includes generalizations of Darboux’s method of finding multivalued first integrals, Jouanolou’s application of this method to the computation of rational first integrals (Theorem 3.1, p. 94) and the algorithm of Prelle-Singer (Theorem 3.9, p. 131).

The first section of chapter 4 is devoted to a generalization of the question posed by Poincaré in the paragraph quoted above, which became known as Poincaré’s Problem:

Given a system of differential equations with polynomial coefficients, find an upper bound on the degree of its irreducible Darboux polynomials.

However, most of the results in this direction are concerned with systems that are not algebraically integrable because, by Jouanolou’s result, such systems have only finitely many irreducible Darboux polynomials. Although this problem is still unsolved, several special cases have been successfully dealt with. In his monograph Jouanolou gave an upper bound on the degree of those Darboux polynomials that define smooth curves [1, Proposition 4.1, p. 126]. In a paper [2] published in 1991 D. Cerveau and A. Lins Neto showed that the same bound holds when the only singularities of the curve defined by the Darboux polynomial are nodes. From an effective point of view, this solution is more interesting than Jouanolou’s because it is possible to impose conditions on the differential system that force its Darboux polynomials to define nodal curves. An even more general result in the same direction, proved by M. Carnicer in [3], is discussed in some detail in subsection 4.1.3. The second section of chapter 4 includes a characterization of Darboux polynomials for the Liénard and Lorenz systems.

Chapter 5 is mostly concerned with the integrability of Hamiltonian systems and other systems of interest in physics. Bruns’s well-known result on algebraic integrals is presented in section 5.1, which also includes a discussion of the 5-dimensional Lorenz equation and of two systems that model the motion of rigid bodies. Section 5.3 includes a brief presentation of the Moralis-Ramis method, based on differential Galois theory, which has been used to prove the non-integrability of several Hamiltonian systems. The first three sections of chapter 6 deal with applications of Darboux polynomials to the centre problem and to the study of limit cycles (Hilbert’s 16th problem), while its final section is a report on the integrability properties of a number of differential systems used to model problems in physics, biology and economics. Finally, chapter 7 is devoted to local integrability questions.

As the author makes clear in the Preface, his aim in writing the book was that of summarizing

the last two decades of research on the integrability of dynamical systems and related topics obtained by [himself] and his co-authors, together with relevant results of other specialists.

This is a fair description of the book, and it is not true that it “can be used as a textbook for graduate students in dynamical systems” as claimed in the text on the back cover. The presentation of the material is somewhat uneven. Although many sections contain complete proofs, there are important results whose proofs are only sketched, while whole sections (like 6.4) are basically surveys of what is known on a given subject.

The author’s choice of focusing on algebraic and analytic methods, and more or less avoiding results from algebraic geometry, sometimes gives the reader a very distorted view of a given result or group of results. Take Carnicer’s solution of the Poincaré Problem, for example. While it is claimed on p. 157 that the main technique in its proof is the notion of blow-up from algebraic geometry, this is defined nowhere, nor does it play any role in the “main ideas and steps” of this proof that are given in pp. 158163. This gives the impression that the dicritical condition a foliation has to satisfy for Carnicer’s bound to hold is non-effective, which is by no means the case. Possibly for the same reason, a number of recent algorithms concerned with rational first integral have been omitted; see [4] and [5], for example.

There are also a number of places where the author did not get things quite right. For example, the homogenization procedure on p. 154 does not take into account the case when the line at infinity is not invariant under the foliation; see [3], for example. He has also misconstrued the statement of a theorem of A. Lins Neto and M. Soares in [6], which is concerned with algebraic invariant curves, not hypersurfaces, as stated in p. 164 (Theorem 4.3). There are also some baffling statements, as when the author says (p. 169) that the proof of a result for which a counter-example has been found is “too technical to be checked”.

On the plus side, the book presents a unique collection of a large number of results concerning Darboux polynomials and their many applications, some of which are quite recent. This is complemented by a bibliography that lists 488 publications. Moreover, even when the proof of a given result is only sketched, enough detail is usually given to enable one to get a good idea of the kind of technique that is required. The book also contains many unsolved problems and a very useful index.


    1. J.P. Jouanolou, Equations de Pfaff algébriques, Lect. Notes in Math. 708, Springer-Verlag, Heidelberg (1979).
  1. D. Cerveau and A. Lins Neto, Holomorphic foliations in \(\mathbf{CP}(2)\) having an invariant algebraic curve, Ann. Sc. de l’Institute Fourier, 41 (1991), 883903.
  2. M.M. Carnicer, The Poincaré problem in the nondicritical case, Ann. Math., 140 (1994), 289—294.
  3. C. Galindo and F. Monserrat, Algebraic integrability of foliations of the plane, J. Differential Equations 231 (2006), no. 2, 611632.
  4. C. Galindo and F. Monserrat, The Poincaré problem, algebraic integrability and dicritical divisors, J. Differential Equations 256 (2014), no. 11, 3614—3633.
  5. A. Lins Neto, A. and M. G. Soares, Algebraic solutions of one-dimensional foliations, J. Differential Geom 43 (1993), 652673.

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