Lectures on analytic differential equations / Yulij Ilyashenko, Sergei Yakovenko.

Format:

Book

Author/Creator:

llyashenko, Yulij, author.

Yakovenko, S., author.

Series:

Graduate studies in mathematics ; Volume 86.

Graduate Studies in Mathematics, 1065-7339 ; Volume 86

Language:

English

Subjects (All):

Differential equations, Nonlinear.

Riemann-Hilbert problems.

Analytic spaces.

Dynamics.

Physical Description:

1 online resource (641 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, 2008.

Language Note:

English

Summary:

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 Poincare 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 mor 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.

Contents:

""Cover""; ""Title""; ""Copyright""; ""Contents""; ""Preface""; ""Chapter I. Normal forms and desingularization""; ""1. Analytic differential equations in the complex domain""; ""2. Holomorphic foliations and their singularities""; ""3. Formal flows and embedding theorem""; ""4. Formal normal forms""; ""5. Holomorphic normal forms""; ""6. Finitely generated groups of conformal germs""; ""7. Holomorphic invariant manifolds""; ""8. Desingularization in the plane""; ""Chapter II. Singular points of planar analytic vector fields""; ""9. Planar vector fields with characteristic trajectories""

""10. Algebraic decidability of local problems and center-focus alternative""""11. Holonomy and first integrals""; ""12. Zeros of parametric families of analytic functions and small amplitude limit cycles""; ""13. Quadratic vector fields and the Bautin theorem""; ""14. Complex separatrices of holomorphic foliations""; ""Chapter III. Local and global theory of linear systems""; ""15. General facts about linear systems""; ""16. Local theory of regular singular points and applications""; ""17. Global theory of linear systems: holomorphic vector bundles and meromorphic connexions""

""18. Riemann�Hilbert problem""""19. Linear nth order differential equations""; ""20. Irregular singularities and the Stokes phenomenon""; ""Appendix: Demonstration of Sibuya theorem""; ""Chapter IV. Functional moduli of analytic classification of resonant germs and their applications""; ""21. Nonlinear Stokes phenomenon for parabolic and resonant germs of holomorphic self-maps""; ""22. Complex saddles and saddle-nodes""; ""23. Nonlinear Riemann�Hilbert problem""; ""24. Nonaccumulation theorem for hyperbolic polycycles""; ""Chapter V. Global properties of complex polynomial foliations""

""25. Algebraic leaves of polynomial foliations on the complex projective plane P[sup(2)]""""Appendix: Foliations with invariant lines and algebraic leaves of foliations from the class A[sup(r)]""; ""26. Perturbations of Hamiltonian vector fields and zeros of Abelian integrals""; ""27. Topological classification of complex linear foliations""; ""28. Global properties of generic polynomial foliations of the complex projective plane P[sup(2)]""; ""First aid""; ""A. Crash course on functions of several complex variables""; ""B. Elements of the theory of Riemann surfaces""; ""Bibliography""

""Index""""A""; ""B""; ""C""; ""D""; ""E""; ""F""; ""G""; ""H""; ""I""; ""J""; ""K""; ""L""; ""M""; ""N""; ""O""; ""P""; ""Q""; ""R""; ""S""; ""T""; ""U""; ""V""; ""W""; ""Back Cover""

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

Description based on print version record.

ISBN:

1-4704-2116-X