Automation and Remote Control

, Volume 72, Issue 5, pp 1036–1047 | Cite as

On perturbations of systems with multidimensional degeneration

  • M. I. Kamenskii
  • B. A. Mikhailenko
Robust and Adaptive Systems

Abstract

Bifurcation conditions are found for the periodic solutions in systems of differential equations with the perturbation (small disturbance) in the case of existence of joined Floke solutions in a linearized nonperturbed system. For this case a multidimensional analog of the Malkin bifurcation function is built up.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Malkin, I.G., Nekotorye zadachi teorii nelineinykh kolebanii (Some Problems of the Theory of Nonlinear Oscillations), Moscow: Gos. Izd. Tekh. Teor. Liter., 1956.Google Scholar
  2. 2.
    Loud, W.S., Periodic Solutions of a Perturbed Autonomous System, Ann. Math., 1959, vol. 70, pp. 490–529.MathSciNetCrossRefGoogle Scholar
  3. 3.
    Blekhman, I.I., Sinkhronizatsiya dinamicheskikh system (Synthronization of Dynamic Sistems), Moscow: Nauka, 1971.Google Scholar
  4. 4.
    Krasnosel’skii, M.A., Operator sdviga po traektoriyam differentsial’nykh uravnenii (The Shift Operator along Trajectories of Differential Equations), Moscow: Nauka, 1966.Google Scholar
  5. 5.
    Akhmerov, R.R., Periodic Solutions of Systems of Autonomous Functionally Differential Equations of the Neutral Type with Small Delay, Diff. Uravn., 1974, vol. 10, no. 11, pp. 1923–1931.Google Scholar
  6. 6.
    Aizengendler, P.G., On Bifurcation of Periodic Solutions of Differential Equations with Delay. I, Izv. Vyssh. Uchebn. Zaved., 1969, no. 10, pp. 3–10.Google Scholar
  7. 7.
    Aizengendler, P.G., On Bifurcation of Periodic Solutions of Differential Equations with Delay. II, Izv. Vyssh. Uchebn. Zaved., 1969, no. 11, pp. 3–12.Google Scholar
  8. 8.
    Kamenskii, M.I., On Bifurcation in the Implicit Function Theorem with Uneven Conditions, in Sovremennye metody teorii kraevykh zadach (Modern Methods of the Theory of Boundary-Value Problems), Voronezh: Voronezh. Gos. Univ., 2009, pp. 79–80.Google Scholar
  9. 9.
    Kamenskii, M.I., On Variational Interpretation of the Problem on Bifurcation of Periodic Solution of Differential Equations in the Case of Nonisolated Positions of Equilibrium of the Neutralized Equation, in Voronezh. zimnyaya mat. shk. S.G. Kreina (Krein Voronezh Winter School), Voronezh: Voronezh State Univ., 2010, pp. 74–75.Google Scholar
  10. 10.
    Hartman, P., Ordinary Differential Equations, New York: Wiley, 1964. Translated under the title Obyknovennye differentsial’nye uravneniya, Moscow: Mir, 1970.MATHGoogle Scholar
  11. 11.
    Mery nekompaktnosti i uplotnyaushchie operatory (Measures of Noncompactness and Condensing Operators), Akhmerov, R.R. et al., Eds., Novosibirsk: Nauka, 1986.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2011

Authors and Affiliations

  • M. I. Kamenskii
    • 1
  • B. A. Mikhailenko
    • 1
  1. 1.Voronezh State UniversityVoronezhRussia

Personalised recommendations