Ukrainian Mathematical Journal

, Volume 45, Issue 4, pp 598–608 | Cite as

Asymptotic expansions of solutions to singularly perturbed systems

  • I. N. Shchitov


Under the condition that a degenerate system has an exponentially stable integral manifold, an asymptotic expansion of the Cauchy problem that generalizes the well known Vasil'eva expansion is constructed for a perturbed system.


Cauchy Problem Asymptotic Expansion Integral Manifold Degenerate System Stable Integral Manifold 
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  1. 1.
    A. N. Tikhonov, “Systems of differential equations with small parameters by the derivatives,”Mat. Sb.,31, No. 3, 575–586 (1952).Google Scholar
  2. 2.
    A. B. Vasil'eva and V. F. Butuzov,Asymptotic Expansions of Solutions to Singularly Perturbed Equations [in Russian], Nauka, Moscow (1973).Google Scholar
  3. 3.
    K. V. Zadiraka, “On the nonlocal integral manifold of an irregularly perturbed differential system,”Ukr. Mat. Zh.,17, No. 1, 47–63 (1965).Google Scholar
  4. 4.
    Yu. A. Mitropol'skii and O. B. Lykova,Integral Manifolds in Nonlinear Mechanics [in Russian], Nauka, Moscow (1973).Google Scholar
  5. 5.
    V. V. Strygin and V. A. Sobolev,Separation of Motions by the Integral Manifold Method [in Russian], Nauka, Moscow (1988).Google Scholar
  6. 6.
    I. N. Shchitov, “On a generalization of the A. N. Tikhonov theorem,”Ukr. Mat. Zh.,38, No. 3, 394–397 (1986).Google Scholar
  7. 7.
    I. N. Shchitov, “On the asymptotics of solutions to the Cauchy problem for a singularly perturbed system,”Differents. Uravn.,21, No. 10, 1823–1825 (1985).Google Scholar
  8. 8.
    Yu. I. Neimark, “Integral manifolds of differential equations,”Izv. Vyssh. Ucheb. Zaved., Radiofizika,10, No. 3, 321–334 (1967).Google Scholar
  9. 9.
    N. Fenichel, “Persistence and smoothness of invariant manifolds for flows,”Indiana Univ. Math. J.,21, No. 3, 193–226 (1971).Google Scholar
  10. 10.
    M. Hirsh, C. Pugh, and M. Shub,Invariant Manifolds, Springer, Berlin (1977).Google Scholar
  11. 11.
    A. B. Vasil'eva and V. F. Butuzov,Singularly Perturbed Equations in Critical Cases [in Russian], Moscow University, Moscow (1978).Google Scholar
  12. 12.
    L. S. Pontryagin and L. V. Rodygin, “An approximate solution of a system of ordinary differential equations with a small parameter by the derivatives,”Dokl. Akad. NaukSSSR,131, No. 2, 255–258 (1960).Google Scholar
  13. 13.
    I. N. Shchitov, “The asymptotics of solutions for systems with slow and fast variables,”Ukr. Mat. Zh.,39, No. 5, 631–637 (1987).Google Scholar
  14. 14.
    I. G. Malkin,The Theory of Stability of Motion [in Russian], Nauka, Moscow (1966).Google Scholar
  15. 15.
    V. V. Rumyantsev and A. S. Oziraner,Stability and Stabilization of Motion with Respect to a Part of Variables [in Russian], Nauka, Moscow (1987).Google Scholar
  16. 16.
    Yu. L. Daletskii and M. G. Krein,Stability of Solutions to Differential Equations in the Banach Space [in Russian], Nauka, Moscow (1970).Google Scholar
  17. 17.
    V. M. Alekseev, “On an estimate of perturbations of solutions to ordinary differential equations,”Vestn. Mosk. Univ., Ser. Mat. Mekh., No. 2, 28–36 (1961).Google Scholar

Copyright information

© Plenum Publishing Corporation 1993

Authors and Affiliations

  • I. N. Shchitov
    • 1
  1. 1.Leningrad Institute of Cinema EngineersLeningrad

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