Geometrical Methods in the Theory of Ordinary Differential Equations

Since the first edition of this book, geometrical methods in the theory of ordinary differential equations have become very popular and some progress has been made partly with the help of computers. Much of this progress is represented in this revised, expanded edition, including such topics as the Feigenbaum universality of period doubling, the Zoladec solution, the Iljashenko proof, the Ecalle and Voronin theory, the Varchenko and Hovanski theorems, and the Neistadt theory. In the selection of material for this book, the author explains basic ideas and methods applicable to the study of differential equations. Special efforts were made to keep the basic ideas free from excessive technicalities. Thus the most fundamental questions are considered in great detail, while of the more special and difficult parts of the theory have the character of a survey. Consequently, the reader needs only a general mathematical knowledge to easily follow this text. It is directed to mathematicians, aswell as all users of the theory of differential equations.

1 Special Equations.- § 1. Differential Equations Invariant under Groups of Symmetries.- § 2. Resolution of Singularities of Differential Equations.- § 3. Implicit Equations.- § 4. Normal Form of an Implicit Differential Equation in the Neighborhood of a Regular Singular Point.- § 5. The Stationary Schrödinger Equation.- § 6. Geometry of a Second-Order Differential Equation and Geometry of a Pair of Direction Fields in Three-Dimensional Space.- 2 First-Order Partial Differential Equations.- § 7. Linear and Quasilinear First-Order Partial Differential Equations.- § 8. The Nonlinear First-Order Partial Differential Equation.- § 9. A Theorem of Frobenius.- 3 Structural Stability.- § 10. The Notion of Structural Stability.- §11. Differential Equations on the Torus.- § 12. Analytic Reduction of Analytic Circle Diffeomorphisms to a Rotation.- § 13. Introduction to the Hyperbolic Theory.- § 14. Anosov Systems.- § 15. Structurally Stable Systems Are Not Everywhere Dense.- 4 Perturbation Theory.- § 16. The Averaging Method.- § 17. Averaging in Single-Frequency Systems.- § 18. Averaging in Systems with Several Frequencies.- § 19. Averaging in Hamiltonian Systems.- § 20. Adiabatic Invariants.- § 21. Averaging in Seifert’s Foliation.- 5 Normal Forms.- § 22. Formal Reduction to Linear Normal Forms.- § 23. The Case of Resonance.- § 24. Poincaré and Siegel Domains.- § 25. Normal Form of a Mapping in the Neighborhood of a Fixed Point.- § 26. Normal Form of an Equation with Periodic Coefficients.- § 27. Normal Form of the Neighborhood of an Elliptic Curve.- § 28. Proof of Siegel’s Theorem.- 6 Local Bifurcation Theory.- § 29. Families and Deformations.- § 30. Matrices Depending on Parameters and Singularities of the Decrement Diagram.- §31. Bifurcations of Singular Points of a Vector Field.- § 32. Versal Deformations of Phase Portraits.- § 33. Loss of Stability of an Equilibrium Position.- § 34. Loss of Stability of Self-Sustained Oscillations.- § 35. Versal Deformations of Equivariant Vector Fields on the Plane.- § 36. Metamorphoses of the Topology at Resonances.- § 37. Classification of Singular Points.- Samples of Examination Problems.