## Vladimir I. Arnold - Collected Works: Hydrodynamics, Bifurcation Theory, and Algebraic Geometry 1965-1972Vladimir Arnold was one of the great mathematical scientists of our time. He is famous for both the breadth and the depth of his work. At the same time he is one of the most prolific and outstanding mathematical authors. This second volume of his Collected Works focuses on hydrodynamics, bifurcation theory, and algebraic geometry. |

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### Contents

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4 Conditions for nonlinear stability of stationary plane curvilinear flows of an ideal fluid | 21 |

5 On the topology of threedimensional steady flows of an ideal fluid | 26 |

6 On an a priori estimate in the theory of hydrodynamical stability | 30 |

7 On the differential geometry of infinitedimensional Lie groups and its applications to the hydrodynamics of perfect fluids | 33 |

8 On a variational principle for the steady flows of perfect fluids and its applications to problems of nonlinear stability | 70 |

25 Algebraic unsolvability of the problem of Lyapunov stability and the problem of topological classification of singular points | 231 |

26 On the arrangement of ovals of real plane algebraic curves involutions of fourdimensional smooth manifolds and the arithmetic of integral quadra... | 239 |

27 The topology of real algebraic curves works of IG Petrovsky and their development | 250 |

28 On matrices depending on parameters | 255 |

29 Lectures on bifurcations in versal families | 270 |

30 Versal families and bifurcations of differential equations Russian | 340 |

31 Remarks on the behaviour of a flow of a threedimensional i fluid in the presence of a small perturbation of the initial vector field | 349 |

32 The asymptotic Hopf invariant and its applications | 357 |

9 On a characteristic class arising in quantization conditions | 85 |

10 A note on the Weierstrass preparation theorem | 99 |

11 The stability problem and ergodic properties for classical dynamical systems | 106 |

12 A remark on the ramification of hyperelliptic integrals as functions of parameters | 115 |

13 Singularities of smooth mappings | 119 |

14 Remarks on singularities of finite codimension in complex dynamical systems | 162 |

15 Braids of algebraic functions and the cohomology of swallowtails | 170 |

16 Hamiltonian nature of the Euler equations in the dynamics of a rigid body and of an ideal fluid | 173 |

17 On the onedimensional cohomology of the Lie algebra of divergencefree vector fields and rotation numbers of dynamical systems | 177 |

18 The cohomology ring of the colored braid group | 181 |

19 On cohomology classes of algebraic functions invariant under Tschirnhausen transformations | 187 |

20 Trivial problems | 191 |

21 Local problems of analysis | 192 |

22 Algebraic unsolvability of the problem of stability and the problem of topological classification of singular points of analytic systems of differenti... | 197 |

23 On some topological invariants of algebraic functions | 199 |

24 Topological invariants of algebraic functions II | 223 |

33 A magnetic field in a moving conducting fluid with YaB Zeldovich AARuzmaikin and DD Sokolov | 376 |

34 A magnetic field in a stationary flow with stretching in a Riemannian manifold with YaB Zeldovich AARuzmaikin and DD Sokolov | 378 |

35 Stationary magnetic field in a periodic flow with YaB Zeldovich AARuzmaikin and DD Sokolov | 383 |

36 Some remarks on the antidynamo theorem | 386 |

37 Evolution of a magnetic field under the action of transfer and diffusion | 396 |

38 The growth of a magnetic field in a threedimensional steady incompressible flow with EIKorkina | 399 |

39 On evolution of a magnetic field under the action of drift and diffusion Russian | 404 |

40 Exponential scattering of trajectories and its hydrodynamical applications | 419 |

41 Kolmogorovs hydrodynamic attractors | 429 |

42 Topological methods in hydrodynamics with BA Khesin | 433 |

43 Translators preface to J Milnors book Morse Theory | 455 |

Selected Works in Three Volumes Vol I New Methods of Celestial Mechanics | 457 |

45 Comments on the paper On a geometric theorem by Henri Poincaré | 462 |

Acknowledgements | 464 |

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### Common terms and phrases

algebraic function Anal analytic asymptotic bifurcation diagram boundary braid group closed codimension coefficients cohomology classes commutative complex consider coordinates corresponding critical points curl curvature curve cycle defined definition denote diffeomorphisms differential equations dimension divergence-free domain dynamical systems eigenvalues equivalent Euler example exponential finite foliation formula geodesic germ given Hamiltonian Hence homeomorphism homology homology groups Hopf invariant hydrodynamics infinite-dimensional integral intersection invariant jets Jordan block k-jets Lagrangian Lemma Lie algebra linear linking number magnetic field manifold mapping f Math matrices Moscow Nauk neighbourhood normal form obtained one-dimensional operator orbit parameters perturbation plane Poincaré polynomial position problem proof proved quadratic form resonance Riemannian Riemannian manifold sequence singular point smooth mappings solution space stability stationary flow submanifold surface symmetric tangent three-dimensional topological torus trajectories Translated two-dimensional V.I. Arnold values variables vector field velocity field versal deformation versal family zero