Differential Equations

, Volume 55, Issue 8, pp 1011–1016 | Cite as

Bifurcation of the Equilibrium of an Oscillator with a Velocity-Dependent Restoring Force under Periodic Perturbations

  • Yu. N. BibikovEmail author
  • V. R. BukatyEmail author
Ordinary Differential Equations


We study the bifurcation of an oscillator whose restoring force depends on the velocity of motion under periodic perturbations. Separation of variables is used to derive a bifurcation equation. To each positive root of this equation, there corresponds an invariant twodimensional torus (a closed trajectory in the case of a time-independent perturbation) shrinking to the equilibrium position as the small parameter tends to zero. The proofs use methods of the Krylov-Bogolyubov theory for the case of periodic perturbations or the implicit function theorem for the case of time-independent perturbations.


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  1. 1.
    Lyapunov, A.M., Study of one of the special cases of the problem of stability of motion, in Sobranie sochinenii (Collected Papers), Moscow-Leningrad: Izd. Akad. Nauk SSSR, 1956}, vol. 2, pp. 272–331.Google Scholar
  2. 2.
    Bibikov, Yu.N. and Savel’eva, A.G., Periodic perturbations of a nonconservative center, Differ. Equations, 2018, vol. 54, no. 3, pp. 295–299.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bibikov, Yu.N., Pliss, V.A., and Trushina, N.V., On stability of zero solution of an essentially nonlinear second-order differential equation, Vestn. St. Petersb. Univ. Math., 2017, vol. 50, no. 3, pp. 235–241.CrossRefGoogle Scholar
  4. 4.
    Hale, J.K., Integral manifolds of perturbed differential systems, Ann. Math., 1961, vol. 73, no. 3, pp. 496–531.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bibikov, Yu.N. and Bukaty, V.R., Bifurcation of an oscillatory mode under a periodic perturbation of a special oscillator, Differ. Equations, 2019, vol. 55, no. 6, pp. 753–757.CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySt. PetersburgRussia

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