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Geometrical Methods in the Theory of Ordinary Differential Equations

by Arnol'd, V. I. A

  • ISBN: 9780387966496
  • ISBN10: 0387966498

Geometrical Methods in the Theory of Ordinary Differential Equations

by Arnol'd, V. I. A

  • Binding: Hardcover
  • Edition: 0002
  • Publisher: Springer-Verlag
  • Publish date: 06/01/1988
  • ISBN: 9780387966496
  • ISBN10: 0387966498
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Description: 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 Schrdinger 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. Poincar 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.
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