Author | Arnold, V. I. author |
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Title | Geometrical Methods in the Theory of Ordinary Differential Equations [electronic resource] / by V. I. Arnold |
Imprint | New York, NY : Springer New York : Imprint: Springer, 1988 |
Edition | Second Edition |
Connect to | http://dx.doi.org/10.1007/978-1-4612-1037-5 |
Descript | XIII, 351 p. online resource |
1 Special Equations -- ยง 1. Differential Equations Invariant under Groups of Symmetries -- ยง 2. Resolution of Singularities of Differential Equations -- ยง 3. Implicit Equations -- ยง 4. Normal Form of an Implicit Differential Equation in the Neighborhood of a Regular Singular Point -- ยง 5. The Stationary Schrรถdinger Equation -- ยง 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 -- ยง 7. Linear and Quasilinear First-Order Partial Differential Equations -- ยง 8. The Nonlinear First-Order Partial Differential Equation -- ยง 9. A Theorem of Frobenius -- 3 Structural Stability -- ยง 10. The Notion of Structural Stability -- ยง11. Differential Equations on the Torus -- ยง 12. Analytic Reduction of Analytic Circle Diffeomorphisms to a Rotation -- ยง 13. Introduction to the Hyperbolic Theory -- ยง 14. Anosov Systems -- ยง 15. Structurally Stable Systems Are Not Everywhere Dense -- 4 Perturbation Theory -- ยง 16. The Averaging Method -- ยง 17. Averaging in Single-Frequency Systems -- ยง 18. Averaging in Systems with Several Frequencies -- ยง 19. Averaging in Hamiltonian Systems -- ยง 20. Adiabatic Invariants -- ยง 21. Averaging in Seifertโs Foliation -- 5 Normal Forms -- ยง 22. Formal Reduction to Linear Normal Forms -- ยง 23. The Case of Resonance -- ยง 24. Poincarรฉ and Siegel Domains -- ยง 25. Normal Form of a Mapping in the Neighborhood of a Fixed Point -- ยง 26. Normal Form of an Equation with Periodic Coefficients -- ยง 27. Normal Form of the Neighborhood of an Elliptic Curve -- ยง 28. Proof of Siegelโs Theorem -- 6 Local Bifurcation Theory -- ยง 29. Families and Deformations -- ยง 30. Matrices Depending on Parameters and Singularities of the Decrement Diagram -- ยง31. Bifurcations of Singular Points of a Vector Field -- ยง 32. Versal Deformations of Phase Portraits -- ยง 33. Loss of Stability of an Equilibrium Position -- ยง 34. Loss of Stability of Self-Sustained Oscillations -- ยง 35. Versal Deformations of Equivariant Vector Fields on the Plane -- ยง 36. Metamorphoses of the Topology at Resonances -- ยง 37. Classification of Singular Points -- Samples of Examination Problems