The book of Yu. Ilyashenko and S. Yakovenko is devoted to ordinary differential equations in the complex domain. They are of the form \(\dot{x} =F(x,t;\lambda)\), where \(x\) takes values in a complex manifold \(M\), and the time \(t\) as well as possible parameters, denoted by \(\lambda ,\) are complex. Thus the trajectories (solutions) are Riemann surfaces in the phase space \(M\). This area of mathematics is important and very active.
In the Preface the authors write the following: “When the work of this book was essentially over, another similar treatise appeared. In 2006 Henryk Żołądek published the fundamental monograph titled very tellingly ‘The monodromy group’. The scope of both books is surprisingly similar, though the symmetric difference is also very large.” This is true, but I can say that the book [H. Żołądek, The monodromy group. Monografie Matematyczne. Instytut Matematyczny PAN (New Series) 67. Basel: Birkhäuser (2006; Zbl 1103.32015)] covers a much larger area (including differential Galois theory, Hodge structures in algebraic geometry and hypergeometric functions), while Ilyashenko-Yakovenko’s book treats many subjects with greater detail. Also the authors put series of interesting exercises and problems at end of each section.
The book consists of five chapters. The first chapter is about normal forms and desingularization and deals with the local situation.
Recall that two holomorphic vector fields are equivalent if there exists a diffeomorphism (holomorphic or finitely smooth or formal) which conjugates the flows generated by these vector fields, and they are orbitally equivalent if the flows become equivalent after a change of time of one of the flows.
The fundamental theorem about the classification of germs of vector fields with a singular point
\[ \dot{x}=Ax+\dots,\quad x\in(\mathbb{C}^{n},0), \]
is the Poincaré-Dulac theorem which says that only resonant terms (with respect to the eigenvalues of the linear part \(A\)) can remain in this normal form (which is only formal). An analogous theorem holds true for local diffeomorphisms about a fixed point. In the book also the situations with non-diagonal linear part \(A\) are considered and a suitable theorem due to G. R. Belitskii [Normal forms, invariants and local maps (in Russian). Naukova Dumka, Kiev (1979)] is proved.
The problem of analyticity of the Poincaré-Dulac normal form is delicate. The authors give a rather detailed proof of the analyticity in the Poincaré domain (i.e., when the convex hull in \(\mathbb{C}\) of the eigenvalues is separated from zero). In the (opposite) Siegel domain the normalizing series can be divergent. In the case of one-dimensional local diffeomorphisms \(f:z\mapsto e^{2\pi i\alpha }z+\dots \) the analyticity of the linearizing diffeomorphisms depends on whether the irrational number \(\alpha\) satisfies the so-called Brjuno condition: if yes then any such \(f\) is analytically linearizable (A. Brjuno) and otherwise there exist \(f\)’s with the above linear part which are non-linearizable (J.-C. Yoccoz). The reader will not find in the book the proofs of the Brjuno and Yoccoz theorems, but he can find a result about certain dichotomy for parametric systems \(\dot{x}=Ax+\lambda f(x):\) either the convergence of the linearizing series holds for any parameter \(\lambda \in \mathbb{C}\) or for almost any \(\lambda\) (in suitable sense) this series is divergent.
One section is devoted to finitely generated groups of local conformal diffeomorphisms \(f:z\mapsto \lambda z+\dots .\) Such groups arise naturally as groups of holonomy transformations in the theory of holomorphic foliations (which are studied in further sections). The authors follow mainly the survey article [P. M. Elizarov, Yu. S. Ilyashenko, A. A. Shcherbakov and S. M. Voronin, in: Nonlinear Stokes phenomena, Adv. Sov. Math. 14, 57–105 (1993; Zbl 1010.32501)]. They present a classification of such groups when they are either abelian or solvable. In the non-solvable case new phenomena arise. For example, a typical orbit of \(z\in \left( \mathbb{C} ,0\right) \) is dense and there are infinitely many (properly defined) limit cycles. Moreover, if two such groups are topologically conjugated, then they are conjugated analytically (rigidity). The latter results are mainly due to A. Shcherbakov and I. Nakai.
A short section is devoted to the Hadamard-Perron theorem about invariant manifolds for hyperbolic singular points of vector fields in the holomorphic case. In dimension two it is the Briot-Bouquet theorem. It is worth to say that in the non-hyperbolic case an invariant manifold can be non-analytic, e.g. the center separatrix for a two-dimensional saddle-node.
The last section of the first chapter contains a theorem about the resolution of singularities of a holomorphic vector field in a plane. This theorem was first stated by I. Bendixson [Acta Math. 24, 1–88 (1900; JFM 31.0328.03)] in the smooth case. Also a smooth version of it was proved by F. Dumortier [J. Differ. Equations 23, 53–106 (1977; Zbl 0346.58002)]. But the right proof (with proper choice of numerical invariants) was given only by A. van den Essen [Lect. Notes Math. 712, 44–59 (1979; Zbl 0418.34008)]. The proof presented by Ilyashenko and Yakovenko relies upon van den Essen’s ideas, but is rather long. The reason is educational; it is preceded by detailed analysis of the blowing-up transformation and by introduction to the theory of divisors and their intersections and multiplicities.
The second chapter is also local, it focuses on singular points of analytic vector fields in the plane. One of the fundamental problems of real planar vector fields is the center-focus problem, i.e., how to check whether a singular point \(x=0\in \mathbb{R}^{2}\) of a vector field \(\dot{x}=V(x)\) without characteristic trajectories (which could tend to \(x=0\) with a definite direction) is asymptotically stable or unstable or is a center. The authors prove that in general the center-focus problem cannot be solved algebraically (with respect to the coefficients in the Taylor expansion of \(V(x)\)), but when the first nonzero homogeneous part of \(V(x)\) is generic, then the center conditions are algebraic with respect to higher order terms in \(V\). It should be underlined that the analytic solvability of this problem was proved recently by N. B. Medvedeva in a rather technical paper [“On the analytic solvability of the problem of distinguishing between center and focus”, Trudy Mat. Inst. Steklov 256, 7–93 (2006)].
A classical theorem of H. Poincaré and A. Lyapunov states that in the center case the (real) vector field of the form \(\dot{x}_{1}=-x_{2}+\dots\), \(\dot{x}_{2}=x_{1}+\dots\) has a local analytic first integral \(H=x_{1}^{2}+x_{2}^{2}+\dots\). There exists a geometric proof of this result due to R. Moussu. It relies upon the fact that the holonomy map associated with the (complex) separatrix \(x_{1}+ix_{2}+\dots =0\) is the identity. In the book we find this proof as well as its generalization (by J.-F. Mattei and R. Moussu [Ann. Sci. Éc. Norm. Supér. (4) 13, 469–523 (1980; Zbl 0458.32005)]) to the case when a holomorphic vector field in \((\mathbb{C}^{2},0)\) has finite number of separatrices and all other phase curves are relatively closed.
Parallel with the center-focus problem is the problem of cyclicity of such singular points. So we ask how many limit cycles can bifurcate from a focus (or center) after perturbation. To be precise, one considers some natural families \(\dot{x}=F(x;\lambda)\), \(\lambda \in U\subset \mathbb{R}^{k}\), of vector fields which depend analytically on \(x\in \mathbb{R}^{2}\) and \(\lambda\). The Lyapunov-Poincaré focus quantities are analytic functions of \(\lambda \) and define the so-called Bautin ideal in the ring of analytic functions in \(U\). The authors study relations between the properties of the Bautin ideal (like the radicality) and the cyclicity of the singular point. Special attention is devoted to the quadratic case, where the parameters \(\lambda \) correspond to the coefficients in the homogeneous part of \(F\) of second degree; here, the maximal cyclicity is three (N. Bautin) and this follows from the fact that the Bautin ideal is radical in the ring of polynomials in \(\lambda \) invariant with respect to some natural action of \(\mathbb{C}^{\ast}\).
The last section of the second chapter is devoted to the Camacho-Sad theorem about existence of at least one analytic separatrix for a singular point of a holomorphic vector field in dimension two. The authors use a proof due to J. Cano [Proc. Am. Math. Soc. 125, No. 9, 2649–2650 (1997; Zbl 0890.34001)]. The reader will find also more precise relations (due to C. Camacho, A. Lins Neto and P. Sad) between invariants of the local foliation \(\mathcal{F}\) defined by the vector field and invariants of the local invariant analytic curves.
The third chapter is devoted to linear differential systems \(\dot{x}=A(t)x,\) \(x\in \mathbb{C}^{n},\) which are treated from the local point of view as well as from the global one. After introducing some general notions (like the monodromy maps and gauge equivalence of two such systems) and their properties, the authors present a local classification of these systems near a regular singular point \(t=0\), i.e., when the solutions \(x=\varphi (t)\) satisfy \(|\varphi (t)|<C|t|^{-N}\).
More interesting is the global theory. Recall that one of the problems from the Hilbert’s list (the 21\(^{\text{st}}\), known also as the Riemann-Hilbert problem) is to construct a differential system \(\dot{x}=A(t)x,\) \(t\in \mathbb{C},\) with meromorphic matrix valued function \(A(t)\) with given poles \(t_{1},\dots ,t_{m}\) (including \(t=\infty\)) and such that there are given the monodromy matrices \(M_{j},\) \(j=1,\dots ,m,\) corresponding to analytic prolongation of solutions along paths which run around the singular points \(t_{j}.\) It was supposed that the solution to this problem should be looked for in the class of differential systems with regular singular points. A positive solution to this version of the problem was found independently by H. Röhrl [Math. Ann. 133, 1–25; Corrigendum ibid. 472 (1957; Zbl 0088.06001)] and J. Plemelj [Problems in the sense of Riemann and Klein. Interscience Tracts in Pure and Applied Mathematics. 16. New York etc.: Interscience Publishers (1964; Zbl 0124.28203)]. Ilyashenko and Yakovenko present the Röhrl’s proof and use the opportunity to give a kind of course on holomorphic vector bundles and meromorphic connections (together with the geometrical proof of the Birkhoff-Grothendieck theorem about splitting of bundles over \(\mathbb{P}^{1}\)).
Riemann, when he stated this problem, wrote about Fuchsian systems, i.e., when the matrix \(A(t)=\sum A_{j}/(t-t_{j})\) with constant matrices \(A_{j}.\) In the book we find a theorem of A. Bolibruch and V. Kostov about positive solution to the Riemann-Hilbert problem in the Fuchsian class in the case when the system \((M_{1},\dots ,M_{m})\) of monodromy matrices is irreducible (there is no common nontrivial invariant subspace). It was a kind of surprise when A. Bolibruch constructed an example of a meromorphic linear system with regular singular points such that there does not exist any Fuchsian system with the same monodromy matrices as for the Bolibruch’s system. Bolibruch’s example is presented in [D. V. Anosov and A. A. Bolibruch, The Riemann-Hilbert problem: A publication from the Steklov Institute of Mathematics. Aspects of Mathematics. 22. Braunschweig: Vieweg (1994; Zbl 0801.34002); The monodromy group (2006; Zbl 1103.32015)], I also described Bolibruch’s result following his book. But in the book of Ilyashenko and Yakovenko we find a clearer approach to the subject.
We say that a system \((M_{1},\dots ,M_{m})\) of \(n\times n\) matrices is of class B if any matrix \(M_{i}\) is equivalent to one Jordan cell (with eigenvalue \(\lambda _{i}\)) and all the matrices \(M_{j}\) have a common nontrivial invariant subspace (reducibility). If such a system arises as a system of monodromy matrices then it should satisfy the relation \(M_{1}\dots M_{m}=I\). In the case of Fuchsian system (without pole at \(t=\infty\)) the latter relation corresponds to \(\sum A_{j}=0.\) But it can be proved that in the Fuchsian case with the system of monodromy matrices in class B we have a stronger relation: \(\prod \lambda _{i}=1\). An explicit example of a system of three \(4\times 4\) matrices in class B with eigenvalues \(1,1,-1\) provides a counterexample to the Riemann-Hilbert problem in the class of Fuchsian systems.
One section is devoted to higher order linear differential equations
\[ x^{(n)}+ a_{1}(t)x^{(n-1)}+\dots +a_{n}(t)x=0 \]
with meromorphic coefficients \(a_{j}(t)\). Recall that here the notions of regularity and of the Fuchsian property of a singular point coincide. The authors focus their attention on the geometry of jet bundles associated with such equations.
Meromorphic linear systems with irregular singular points are equally important as systems with regular singularities. A typical such system has the form \(t^{r}\dot{x}=B(t)x,\) where \(r>1\) and the matrix \(B(0)\) has distinct eigenvalues. There exists a well defined formal gauge change which reduces this system to a diagonal and polynomial form \(t^{r}\dot{y}=D(t)y\), but this change is non-analytic in general. The reason for the non-analyticity lies in so-called Stokes phenomenon: basic formal solutions are well defined in sectors about \(t=0,\) but the constants before these solutions change when we change the sector. This change is described by means of so-called Stokes operators, which form a kind of Cech 1-cocycle of some sheaf of matrix valued functions. The fundamental theorem of Y. Sibuya (see [Linear differential equations in the complex domain: problems of analytic continuation. Translations of Mathematical Monographs 82. Providence, RI: American Mathematical Society (1990; Zbl 1145.34378)] and B. Malgrange [Lect. Notes Math. 712, 77–86 (1979; Zbl 0423.32014)]) says that the cohomology class of this cocycle is a complete analytic invariant of the singular point with fixed formal normal form. In the book all this is thoroughly presented.
The fourth chapter of the monograph deals with nonlinear complex differential equations.
Two sections are devoted to the nonlinear version of the Stokes phenomena and follow mainly the book [Yu. S. Ilyashenko (ed.), Nonlinear Stokes phenomena. Transl. ed. by A.B. Sossinsky. Advances in Soviet Mathematics. 14. Providence, RI: American Mathematical Society (1993; Zbl 0772.00012)]. Therefore, we find the Ecalle-Voronin moduli of analytic classification (with respect to conjugation in the group \(\text{Diff}(\mathbb{C},0)\) of germs of analytic diffeomorphisms) of conformal maps of the form \(z\mapsto f(z)=z+z^{p+1}+\lambda z^{2p+1}+\dots\). Like in the linear case these moduli arise from the fact that such germs can be reduced to \(f_{0}=z+z^{p+1}+\lambda z^{2p+1}\) only in sectors and the 1-cocycle arising from ‘differences’ between conjugating diffeomorphisms in sectors gives the Ecalle-Voronin modulus.
A variant of Ecalle-Voronin modulus classifies germs like \(f(z)=e^{2\pi ip/q}z+\dots\), i.e., with resonant linear part. The latter moduli are applied to analytic classification (with respect to analytic orbital equivalence) of germs of plane vector fields \(\dot{x}_{1}=px_{1}+\dots\), \(\dot x_{2}=-qx_{2}+\dots \) with resonant singular points. It turns out that the monodromy diffeomorphism associated with one of the separatrices has resonant linear part and its Ecalle-Voronin invariant works well also for the classification of vector fields.
Slightly more complicated invariants are the so-called Martinet-Ramis moduli of analytic orbital classification of saddle-nodes \(\dot{x} _{1}=x_{1}+\dots ,\) \(\dot{x}_{2}=x_{2}^{p+1}+\dots \). Some of these moduli are responsible for possible non-analyticity of the center manifold of the saddle-node.
A section devoted to so-called nonlinear Riemann-Hilbert problem is very interesting. It deals with vector fields in \(\mathbb{P}\times ( \mathbb{C}^{n},0)\) which can be written as a nonlinear non-autonomous system \(\frac{dx}{dt}=A(t)x+O(x^{2})\) with singular points \(\left( t_{j},0\right)\), \(j=1,\dots m.\) Analytic prolongation of solutions of this system along loops in \(\mathbb{P}\), which surround the poles \(t_{j},\) gives a system \(g_{1},\dots ,g_{m}\in Diff(\mathbb{C}^{n},0)\) of holonomy maps, which are subject to the relation \(g_{1}\circ \dots \circ g_{m}=id\). As in the linear Riemann-Hilbert problem one has a system \((g_{1},\dots,g_{m})\) of local diffeomorphisms and asks whether there exists a vector field like above such that the corresponding system of its holonomy maps is \((g_{1},\dots ,g_{m})\).
In the book the answer is given in the case \(n=1,\) i.e., for \(g_{j}(x)=\nu _{j}x+\dots\). Some subtleties arise. For instance, if \(|\nu_{j}|=1\) and the germ \(g_{j}\) is not linearizable, then this germ can arise only as a holonomy map of a saddle and hence \(\nu _{j}=e^{2\pi i\lambda _{j}}\) with \(\lambda _{j}<0\); one defines \(\lambda _{j}=\ln ^{-}\nu _{j}\) if \(-1\leq \lambda _{j}<0\). The answer to the nonlinear Riemann-Hilbert problem for \(n=1\) is positive if: either (1) at least one germ is linearizable, or (2) the collection contains \(k\) germs which can be realized as holonomy maps for resonant nodes and \( k+\sum_{j=0}^{m}\ln ^{-}\nu _{j}\geq -1.\)
Among non-elementary singular points of planar vector fields (i.e., with zero eigenvalues) most important are the so-called Bogdanov-Takens singularities with nilpotent linear part. It was proved in [E. Stróżyna and H. Żołądek, J. Differ. Equations 179, No. 2, 479–537 (2002; Zbl 1005.34034)] that the classical Takens normal form \(\dot{x}_{1}=x_{2}+a(x_{1}),\) \(\dot{x}_{2}=b(x_{1})\) can be obtained by means of an analytic equivalence. The proof uses direct estimates in certain iteration process, but later a geometrical proof was found by F. Loray [Ann. Math. (2) 163, No. 2, 709–722 (2006; Zbl 1103.32018)]. Ilyashenko and Yakovenko reproduce the latter proof in form of series of exercises.
But probably the most important result (obtained independently by Yu. Ilyashenko and J. Ecalle) presented in this chapter is the non-accumulation theorem for limit cycles of analytic vector fields. This theorem is necessary for the proof of finitness of the number of limit cycles of a polynomial real planar vector field. Recall that the famous Hilbert’s 16\(^{ \text{th}}\) problem is to find a bound (i.e., in terms of the degree of the vector field) for the number of limit cycles of such a vector field. After the Ilyashenko-Ecalle theorem, the next aim is to prove the existence of a bound for the number of limit cycles in any local family of polynomial vector fields. The complete proof of the non-accumulation theorem is given in [{Yu. S. Ilyashenko}, Finiteness theorems for limit cycles. Transl. from the Russian by H. H. McFaden. Translations of Mathematical Monographs. 94. Providence, RI: American Mathematical Society (1991; Zbl 0743.34036)]. In this chapter, the authors present the proof for limit cycles near a polycycle consisting of hyperbolic saddles and connecting them separatrices.
The last chapter of the book is devoted to holomorphic foliations \(\mathcal{F}\) of the projective plane \(\mathbb{P}^{2}.\) In any affine part it is a foliation of the plane \(\mathbb{C}^{2}\) into phase curves of a polynomial vector field (or a polynomial Pfaff form). There exist interesting questions about such foliations.
One of them was asked by Poincaré and concerns invariant algebraic curves. How many isolated such curves can exist and of how big degree? It turns out that there does not exist a bound for degree if an invariant algebraic curve \(C\) in terms of (suitably defined) degree of the foliation. But in the case when all the singular points of the foliation are not dicritical (i.e., have only finite number of local analytic invariant curves through them) we have \(\deg C\leq \deg \mathcal{F}+1\) [M. M. Carnicer, Ann. Math. (2) 140, No. 2, 289–294 (1994; Zbl 0821.32026)]. On the other hand, we have the example \(\dot{x}_{0}=x_{1}^{s}\), \(\dot{x}_{1}=x_{2}^{s}\), \(\dot{x}_{2}=x_{0}^{s}\), \(s\geq 2\) (due to J.-P. Jouanolou), without invariant algebraic curves (also at infinity).
Another question concerns the existence of a first integral, expressed in some reasonable terms, for a polynomial vector field. G. Darboux observed that if there exist many invariant algebraic curves \(f_{j}(x)=0\), then we have a first integral of the Darboux form \(H=\prod f_{j}^{a_{j}}(x).\)
An infinitesimal version of Hilbert’s 16\(^{\text{th}}\) problem concerns limit cycles born from ovals \(\gamma (h)\subset \{H=h\}\) in polynomial perturbations \(dH+\varepsilon \omega =0\) (in the Pfaff form) of a Hamiltonian foliation. The problem is reduced to the problem of zeroes of the Abelian integrals \(I(h)=\int_{\gamma (h)}\omega .\) The authors present various tools developed in this topics: relative cohomology and the Françoise algorithm, as well as classical methods like Picard-Lefschetz formula, Gelfand-Leray residuum and Picard-Fuchs equations.
One section is devoted to classification of linear foliations in \(\mathbb{C} ^{n}\) with respect to topological orbital equivalence. In the real and hyperbolic case the corresponding result is the Grobman-Hartman theorem. In the complex case the authors introduce a suitable notion of hyperbolicity and prove the following result (among many others):
Let two hyperbolic linear vector fields be defined by diagonal matrices with the eigenvalues \(\lambda _{j}\) and \(\lambda _{j}^{\prime }\) respectively. Introduce two tuples \((T_{1},\dots ,T_{n})\) and \((T_{1}^{\prime },\dots ,T_{n}^{\prime })\), where the periods \(T_{j}=2\pi i/\lambda _{j}\) and \(T_{j}^{\prime }=2\pi i/\lambda _{j}^{\prime}\). Then the two corresponding foliations are topologically equivalent iff the two systems of periods are transformed one to another by means of one \(\mathbb{R}\)-linear map of the complex plane.
In the final section of this chapter the reader will find a comparison between the geometry of real and complex foliations of \(\mathbb{P}_{\mathbb{R}}^{2}\) and of \(\mathbb{P}_{\mathbb{C}}^{2}\) defined by means of polynomial vector fields. It turns out that leaves of generic foliation are dense in \(\mathbb{P}_{\mathbb{C}}^{2}\) and the foliation has infinitely many (suitably defined) limit cycles. Moreover the complex foliations exhibit the following rigidity property: if two generic foliations are topologically conjugated then they are conjugated by means of a projective automorphism of the projective plane.
We see that the “Lectures on analytic differential equations” is a highly valuable book. It will be useful for students which enter into the area of differential equation (in a wide sense) as well as for experienced mathematicians interested in this subject.