Hopf bifurcation in presence of 1 : 3 resonance

Anna Cima, Jaume Libre, Marco Antonio Teixeira

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1 Citation (Scopus)


Assume that the linear part at a singular point p0of a C4differential system Y0in ℝ4has eigenvalues ±αi and ±βi such that β/α = 1/3. In the main result of the paper we exhibit a one-parameter family of systems Yεfor ε ∈ (-δ0, +δ0) where is shown that the original vector field around p0can bifurcate in 0, 1, 2, 3 or 4 one-parameter families of periodic orbits. The tool for proving such a result is the averaging theory for non-C1differentiable system. Moreover, assuming now that Yεis a one-parameter family of ℤ2-reversible polynomial vector fields of degree 5, we show that it can bifurcate in 0 or 2 one-parameter families of periodic orbits.
Original languageEnglish
Pages (from-to)619-632
JournalAdvanced Nonlinear Studies
Issue number3
Publication statusPublished - 1 Jan 2008


  • Averaging theory
  • Hopf bifurcation
  • Liapunov center theorem
  • Limit cycle
  • Periodic orbit


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