Journal of Dynamics and Differential Equations

, Volume 23, Issue 3, pp 425–435

An Alternative Approach to Study Bifurcation from a Limit Cycle in Periodically Perturbed Autonomous Systems


DOI: 10.1007/s10884-011-9207-4

Cite this article as:
Kamenskii, M., Makarenkov, O. & Nistri, P. J Dyn Diff Equat (2011) 23: 425. doi:10.1007/s10884-011-9207-4


The goal of this paper is to present a new method to prove bifurcation of a branch of asymptotically stable periodic solutions of a T-periodically perturbed autonomous system from a T-periodic limit cycle of the autonomous unperturbed system. The method is based on a linear scaling of the state variables to convert, under suitable conditions, the singular Poincaré map (with two singularity conditions) associated to the perturbed autonomous system into an equivalent non-singular equation to which the classical implicit function theorem applies directly. As a result we obtain the existence of a unique branch of T-periodic solutions (usually found for bifurcations of co-dimension 2) as well as a relevant property of the spectrum of their derivatives. Finally, by a suitable representation formula of the classical Malkin bifurcation function, we show that our conditions are equivalent to the existence of a non-degenerate simple zero of the Malkin function. The novelty of the method is that it permits to solve the problem without explicit reduction of the dimension of the state space as it is usually done in the literature by the Lyapunov–Schmidt method.


Bifurcation Poincaré map Limit cycle Periodic solutions Stability 

Mathematics Subject Classification (2000)

37G15 34E10 34C25 

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Mikhail Kamenskii
    • 1
  • Oleg Makarenkov
    • 2
  • Paolo Nistri
    • 3
  1. 1.Department of MathematicsVoronezh State UniversityVoronezhRussia
  2. 2.Department of MathematicsImperial College LondonLondonUK
  3. 3.Dipartimento di Ingegneria dell’InformazioneUniversità di SienaSienaItaly

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