Advertisement

Singular Problems

  • Valery V. Kozlov
  • Stanislav D. Furta
Chapter
  • 1.2k Downloads
Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

The title of this section scarcely differs from the title of the first section of the preceding chapter. However, the present section is dedicated to the problem of the precise influence of the zero roots on the stability of a critical point and on the existence of asymptotic solutions when there exist roots of the characteristic equation of first approximation system that are different from zero.

Keywords

Asymptotic Solution Invariant Manifold Formal Power Series Center Manifold Negative Real Part 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Literature

  1. 2.
    Appel’rot, G.G.: On §2 of S.V. Kovalevsky’s memoir Sur le problème de la rotation d’un corps solide autour d’un point fixe. Mat. Sb. 16(1), 483–507, 592–596 (1892)Google Scholar
  2. 7.
    Arnold, V.I.: Geometrical Methods in the Theory of Ordinary Differential Equations. Springer, New York (1988). Publication source: Nauka, Moscow (1978)Google Scholar
  3. 18.
    Bogolyubov, N.N., Mitropolsky, Y.O.: Asymptotic Methods in the Theory of Non-linear Oscillations. Gordon and Breach, New York (1961)zbMATHGoogle Scholar
  4. 20.
    Bogoyavlenskiy, A.A.: On some special solutions of the problem of motion of a heavy rigid body about a fixed point. J. Appl. Math. Mech. 22, 1049–1064 (1958)MathSciNetCrossRefGoogle Scholar
  5. 21.
    Bogoyavlenskiy, A.A.: On particular cases of motion of a heavy rigid body about a fixed point. J. Appl. Math. Mech. 22, 873–906 (1958)MathSciNetCrossRefGoogle Scholar
  6. 25.
    Bolotin, S., Negrini, P.: Asymptotic solutions of Lagrangian systems with gyroscopic forces. Nonlinear Differ. Equ. Appl. 2(4), 417–444 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 27.
    Bountis, T.C., Drossos, L.B., Lakshmanan, M., Parthasarathy, S.: On the non-integrability of a family of Duffing – van der Pol oscillators. J. Phys. A 26(23), 6927–6942 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 36.
    Chang, Y.F., Tabor, M., Weiss, J.: Analytic structure of the Henon-Heiles Hamiltonian in integrable and nonintegrable regimes. J. Math. Phys. 23(4), 531–538 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 37.
    Chang, Y.F., Greene, J.M., Tabor, M., Weiss, J.: The analytic structure of dynamical systems and self-similar natural boundaries. Phys. D 8(1–2), 183–207 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 41.
    Chow, S.-N., Hale, J.K.: Methods of Bifurcation Theory. Springer, New York (1982)zbMATHCrossRefGoogle Scholar
  11. 52.
    Fournier, J.D., Levine, G., Tabor, M.: Singularity clustering in the Duffing oscillator. J. Phys. A 21, 33–54 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 59.
    Furta, S.D.: Asymptotic solutions of autonomous systems of ordinary differential equations in the critical case of several zero eigenvalues (Russian). In: Problems in the Mechanics of Controlled Motion. Nonlinear Dynamical Systems, pp. 152–156. PGU, Perm’ (1989)Google Scholar
  13. 66.
    Gantmacher, F.R.: The Theory of Matrices, vols. 1, 2. Chelsea, New York (1959)Google Scholar
  14. 70.
    Grodzenskiy, S.Y.: Andrey Andreyevich Markov (Russian), pp. 1865–1922. Nauka, Moscow (1987). MR 909492Google Scholar
  15. 75.
    Hardy, G.H.: Divergent Series. Clarendon, Oxford (1949)zbMATHGoogle Scholar
  16. 79.
    Hess, W.: Über die Euler’schen Bewegungsgleichungen und über eine neue particuläre Lösung des Problems der Bewegung eines starren Körpers um einen festen Punkt. Math. Annal. 37(2), 153–181 (1890)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 83.
    Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge/New York (1991)zbMATHCrossRefGoogle Scholar
  18. 102.
    Kovalevsky, S.: Sur le problème de la rotation d’un corps solide autour d’un point fixé, I,II. Acta Math. 12, 177–232 (1889); 14, 81–93 (1890)Google Scholar
  19. 115.
    Kozlov, V.V., Furta, S.D.: On solutions with generalized power asymptotics to systems of differential equations. Math. Notes 58(6), 1286–1293 (1995). Translated from: Mat. Zametki 58(6), 851–861, 959 (1995)Google Scholar
  20. 118.
    Kozlov, V.V., Treshchev, D.V.: Kovalevskaya numbers of generalized Toda chains. Math. Notes 46(5), 840–848 (1989). Translated from: Mat. Zametki 46(5), 17–28 (1989)Google Scholar
  21. 125.
    Kuznetsov, A.N.: Differentiable solutions to degenerate systems of ordinary equations. Funct. Anal. Appl. 6, 119–127 (1972). Translated from: Funkts. Anal. Prilozh. 6(2), 41–51 (1972)Google Scholar
  22. 126.
    Kuznetsov, A.N.: Existence of solutions entering at a singular point of an autonomous system having a formal solution. Funct. Anal. Appl. 23(4), 308–317 (1989). Translated from: Funkts. Anal. Prilozh. 23(4), 63–74 (1989)Google Scholar
  23. 130.
    Lunev, V.V.: Meromorphic solutions of the equations of motion of a heavy solid with a fixed point. J. Appl. Math. Mech. 58(1), 31–41 (1994). Translated from: Prikl. Mat. Mekh. 58(1), 30–39 (1994)Google Scholar
  24. 132.
    Lyapunov, A.M.: On a property of the differential equations for the problem of the motion of a solid body with fixed point (Russian). In: Collected Works, vol. 1, 407–417. Akademii Nauk SSSR, Moscow/Leningrad (1954)Google Scholar
  25. 133.
    Lyapunov, A.M.: The General Problem of the Stability of Motion. Taylor & Francis, London (1992)zbMATHGoogle Scholar
  26. 135.
    Markeev, A.P.: On motions asymptotic to triangular points of libration of the restricted circular three-body problem. J. Appl. Math. Mech. 51(3), 277–282 (1987). Translated from: Prikl. Mat. Mekh. 51(3), 355–362 (1987)Google Scholar
  27. 137.
    Marsden, J.E., McCracken, M.: The Hopf Bifurcation and Its Applications. Springer, New York (1976)zbMATHCrossRefGoogle Scholar
  28. 178.
    Sokol’skiy, A.G.: On the stability of Hamiltonian systems in the case of zero frequencies (Russian). Differ. Uravn. 17(8), 1509–1510 (1981). Zbl 0467.34043Google Scholar
  29. 179.
    Spring, F., Waldvogel, J.: Chaos in coorbital motion. In: Roy, A.E. (ed.) Predictability, Stability and Chaos in N-Body Systems, pp. 395–410. Plenum, New York (1991)Google Scholar
  30. 181.
    Tabor, M., Weiss, J.: Analytic structure of the Lorenz system. Phys. Rev. A 24(4), 2157–2167 (1981)MathSciNetCrossRefGoogle Scholar
  31. 192.
    Wasow, W.: Asymptotic Expansions for Ordinary Differential Equations. Wiley-Interscience, New York (1965)zbMATHGoogle Scholar
  32. 194.
    Wiener, Z.: Instability with two zero frequencies. J. Differ. Equ. 103, 58–67 (1993)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Valery V. Kozlov
    • 1
  • Stanislav D. Furta
    • 2
  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia
  2. 2.Faculty for Innovative and Technological BusinessRussian Academy of National Economy and Public AdministrationMoscowRussia

Personalised recommendations