Lectures on Analytic Differential Equations

Abstract

The book combines the features of a graduate-level textbook with those of a research monograph and survey of the recent results on analysis and geometry of differential equations in the real and complex domain. As a graduate textbook, it includes self-contained, sometimes considerably simplified demonstrations of several fundamental results, which previously appeared only in journal publications (desingularization of planar analytic vector fields, existence of analytic separatrices, positive and negative results on the Riemann-Hilbert problem, Ecalle-Voronin and Martinet-Ramis moduli, solution of the Poincaré problem on the degree of an algebraic separatrix, etc.). As a research monograph, it explores in a systematic way the algebraic decidability of local classification problems, rigidity of holomorphic foliations, etc. Each section ends with a collection of problems, partly intended to help the reader to gain understanding and experience with the material, partly drafting demonstrations of the more recent results surveyed in the text.The exposition of the book is mostly geometric, though the algebraic side of the constructions is also prominently featured. On several occasions the reader is introduced to adjacent areas, such as intersection theory for divisors on the projective plane or geometric theory of holomorphic vector bundles with meromorphic connections. The book provides the reader with the principal tools of the modern theory of analytic differential equations and intends to serve as a standard source for references in this area. ReadershipGraduate students and research mathematicians interested in analysis and geometry of differential equations in real and complex domain. Ordering information: Visit the AMS online bookstore Draft version is available onlineThe draft posted on this site is really outdated, with many inaccuracies, sloppy phrases and even errors. Besides, a number of sections had been added or modified. It sole purpose is to give an idea what appears in the printed book.Do not quote this draft to avoid misleading enumeration of pages, theorems and definitions. Contents PrefaceThe branch of mathematics which deals with ordinary differential equations can be roughly divided into two large parts, qualitative theory of differential equations and the dynamical systems theory. The former mostly deals with systems of differential equations on the plane, the latter concerns multidimensional systems (diffeomorphisms on two-dimensional manifolds and flows in dimension greater than two and up to infinity). The former can be considered as a relatively orderly world, while the latter is the realm of chaos.A key problem, in some sense a paradigm influencing the development of dynamical systems theory from its origins, is the problem of turbulence: how a deterministic nature of a dynamical system can be compatible with its apparently chaotic behavior. This problem was studied by the precursors and founding fathers of the dynamical systems theory: L. Landau, H. Hopf, A. Kolmogorov, V. Arnold, S. Smal...

Gevrey smooth topology is proper to detect normalization under Siegel type small divisor conditions

“…Here we follow the scheme as shown in [3] (pp. 70-72) and [4] (pp. 52-56) by some modifications due to our case.…”

Stokes Phenomenon and Confluence in Non-autonomous Hamiltonian Systems

“…Note that in our notation ± (u, x, 0) consists of a pair of sectoral transformations (u, x, 0) and (u, x, 0); it is a functional cochain using the terminology of [11,18].…”

Structure of leaves and the complex Kupka-Smale property

“…Definitions 5.1-5.5 are from [14]. They are scatted through out the text, so we provide them here for the convenience of the reader.…”

“…So we reserve the word "hyperbolic" to complex hyperbolicity. Strongly invariant manifolds exist [14,Theorem 7.4]. The proof can be easily modified to show that they depend holomorphically on a foliation.…”

“…Notice that a 1-dimensional holomorphic foliation with singular locus of codimension 2 is locally determined by a holomorphic vector field [14].…”