Transversal intersection of separatrices and branching of solutions as obstructions to the existence of an analytic integral in many-dimensional systems. I. Basic result: separatrices of hyperbolic periodic points

• Autores: Sergei A. Dovbysh
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 50, Fasc. 2, 1999, págs. 119-198
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • It is well-known that the existence of transversally intersecting separatrices of hyperbolic periodic solutions leads, in a typical situation, to complicated and irregular dynamics. Therefore, in the case of a two-dimensional mapping or a three-dimensional flow, with this transversality property, there is no non-trivial analytic or meromorphic first integral, i.e., a function constant along each trajectory of the system under consideration. Additional robust conditions are obtained and discussed that guarantee the absence of such an integral in the manydimensional case, regardless of the finite dimension in question (the strongest analytic non-integrability). These conditions guarantee also the absence of any non-trivial analytic one-parameter symmetry group, and, more generally, analytic or meromorphic vector fields generating a local symmetry, i.e., a local phase flow commuting with the system under consideration. Furthermore, the analytic centralizer of the system is discrete in the compact-open topology. A differential-topological structure of the invariant set of “quasi-random motions” is studied for this purpose. The approach used is essentially geometrical. Some related topics are also discussed.

• Referencias bibliográficas
  • P.S. Aleksandrov, Introduction to Set Theory and General Topology, “Nauka”, Moscow, Russian, 1977.
  • V.M. Alekseev, Quasi-random dynamical systems, I, Quasi-random diffeomorphisms, Mat. Sb. 76(1) (1968), 72–134, English transl. in: Math. USSR-Sb....
  • V.M. Alekseev, Final motions in the three-body problem and symbolic dynamics, Uspekhi Mat. Nauk. 36(4) (1981), 161–176, English transl. in:...
  • V.M. Alekseev and M.V. Yakobson, Symbolic dynamics and hyperbolic dynamical systems, Appendix to R. Bowen, Methods of Symbolic Dynamics, Collected...
  • B. Anderson, Diffeomorphisms with discrete centralizer, Topology 15(2) (1976), 143–147.
  • D.V. Anosov, Geodesic Flows on Closed Riemannian Manifolds of Negative Curvature, Trudy Matem. Inst. Steklova AN SSSR, 90, 1967, English transl....
  • L. Antoine, Sur l’homéomorphie de deux figures et de leurs voisinages, J. Math. Pures Appl. 86 (sér. 8, 4)(3–4) (1921), 221–325.
  • V.I. Arnol’d, V.V. Kozlov and A.I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, Itogi nauki i tekhniki, Dinamicheskie...
  • V.I. Arnol’d, A.N. Varchenko and S.M. Gusein-Zade, Singularities of Differentiable Maps, 1, The Classification of Critical Points, Caustics...
  • H.W. Broer and F.M. Tangerman, From a differentiable to a real analytic perturbation theory; applications to the Kupka-Smale theorem, Ergodic...
  • I.U. Bronshtein, Transversality implies structural stability, Dokl. AN SSSR, 257(1) (1981), 265–268, English transl. in: Sov. Math. Dokl....
  • I.U. Bronshtein, Nonautonomous Dynamical Systems, “Shtiintsa”, Kishinev, Russian, 1984.
  • J.W. Cannon, The recognition problem: what is a topological manifold?, Bull. Amer. Math. Soc. 84(5) (1978), 832–866.
  • J.W. Cannon and D.G. Wright, Slippery Cantor sets in $E^n$, Fundamenta Mathematicae, 106(2) (1980), 89–98.
  • H. Cartan, Calcul Différentiel, Hermann, Paris, 1967.
  • R.J. Duvallerman, Decompositions of Manifolds, Pure and Appl. Math., 124, Orlando etc., Academic Press, 1986.
  • S.A. Dovbysh, Transversal intersection of separatrices and non-existence of an analytical integral in multi-dimensional systems, Variational...
  • S.A. Dovbysh, Transversal intersection of separatrices, the structure of the set of quasi-stochastic motions, and the non-existence of an...
  • M.J. Foster, Fibre derivatives and stable manifolds: a note, Bull. London Math. Soc. 8 Part 3(24) (1976), 286–288.
  • B.A. Fuks, Theory of Analytic Functions of Several Complex Variables, Gostekhizdat, Moscow-Leningrad, 1948, English transl. in: Providence,...
  • F.R. Gantmacher, The Theory of Matrices, Gostekhizdat, Moscow, 1953, 4th ed. by “Nauka”, Moscow, 1988, English transl. in: Chelsea, New York,...
  • M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities, Graduate texts in Mathematics, 14, Springer, New York-Heidelberg-Berlin,...
  • H. Grauert, On Levi’s problem and the imbedding of real analytic manifolds, Ann. Math. Ser. 2, 68(2) (1958), 460–472.
  • J. Hadamard, Sur l’itération et les solutions asymptotiques des équations différentielles, Bull. Soc. Math. France 29 (1901), 224–228.
  • J.K. Hale and Xiao-Biao Lin, Symbolic dynamics and nonlinear semiflows, Ann. Math. Pura Appl. 144 (1986), 229–259.
  • P. Hartman, Ordinary Differential Equations, Wiley, New York-London-Sydney, 1964, reprint 2nd ed. by Birkhäuser, Boston-Basel-Stuttgart, 1982.
  • M.W. Hirsch, C.C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Math. 583, Springer, Berlin-Heidelberg-New York, 1977.
  • J. Kajiwara and E. Sakai, Generalization of Levi-Oka’s theorem concerning meromorphic functions, Nagoya Math. J. 29 (1967), 75–84.
  • N. Kopell, Commuting diffeomorphisms, Global Analysis, (S.-S. Chern, S. Smale, eds.), Proc. Symp. in Pure Math., 14, Amer. Math. Soc., Providence...
  • V.V. Kozlov, Integrability and non-integrability in Hamiltonian mechanics, Uspekhi Mat. Nauk 38(1) (1983), 3–67, English transl. in: Russian...
  • V.V. Kozlov, On the symmetry groups of dynamical systems, Prikl. Matem. i Mekhan. 52(4) (1988), 531–541, English transl. in: J. Appl. Math....
  • A.G. Kushnirenko, A.B. Katok and V.M. Alekseev, Smooth dynamical systems, Ninth Summer Mathematical School, “Naukova Dumka”, Kiev, 1976, pp....
  • L.M. Lerman and L.P. Shil’nikov, Homoclinic structures in infinite-dimensional systems, Sibir. Matem. Zh. 29(3) (1988), 92–103, English transl....
  • J. Moser, Stable and Random Motions in Dynamical Systems, With Special Emphasis on Celestial Mechanics, Ann. Math. Stud. 77, Princeton Univ....
  • R. Narasimhan, Analysis on Real and Complex Manifolds, Adv. Stud. in Pure Math., 1 North-Holland, Amsterdam, 1968.
  • V.V. Nemyckii and V.V. Stepanov, Qualitative Theory of Differential Equations, Gostekhizdat, Moscow-Leningrad, 1947, 2nd ed., 1949, English...
  • Z. Nitecki, Differentiable Dynamics. An Introduction to the Orbit Structure of Diffeomorphisms, MIT Press, Cambridge (Mass.)-London, 1971.
  • I. Palis, Rigidity of the centralizers of diffeomorphisms and structural stability of suspended foliations, Differential Topology, Foliations...
  • J. Palis and W. de Melo, Geometric Theory of Dynamical Systems. An Introduction, Springer, New York-Heidelberg-Berlin, 1982.
  • I. Palis and J.C. Yoccoz, Rigidity of centralizers of diffeomorphisms, Ann. Scient. École Norm. Super. 4e série 22(1) (1989), 81–98.
  • J. Palis and J.C. Yoccoz, Centralizers of Anosov diffeomorphisms on tori, ibid, 99–108.
  • H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste, 3, Gauthier-Villars, Paris, 1899, reprinted by Dover Publications, New York,...
  • J. Rocha, Rigidity of centralizers of real analytic diffeomorphisms, Ergodic Theory and Dynamical Systems, 13(1) (1993), 175–197.
  • B.V. Shabat, Introduction to Complex Analysis, 2, Functions of Several Variables, “Nauka”, Moscow, 2nd ed., 1976, English transl. in: Transl....
  • M.A. Shayman, Phase portrait of the matrix Riccati equation, SIAM J. Control and Optimization, 24(1) (1986), 1–65.
  • M.A. Shayman, Riccati equations, linear flows on the flag manifold, and the Bruhat decomposition, Frequency Domain and State Space Methods...
  • S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73(6) (1967), 747–817.
  • S. Sternberg, Local contractions and a theorem of Poincaré, Amer. J. Math. 79(4) (1957), 809–824.
  • Y. Yomdin, Nonautonomous linearization, Dynamical Systems (J.C. Alexander, ed.), Lecture Notes in Math. 1342, Springer, Berlin-Heidelberg-New...