WIAS Preprint No. 1634, (2011)

Andronov--Hopf bifurcation of higher codimensions in a Liénard system


  • Schneider, Klaus
  • Grin, Alexander

2010 Mathematics Subject Classification

  • 34A34 34C05 34C23 37G15

2008 Physics and Astronomy Classification Scheme

  • 05.45.Xt 89.75.Kd


  • Degenerate Hopf bifurcation, codimension three, multiple limit cycles, bifurcation diagrams




Consider a polynominal Liènard system depending on three parameters itshape a, b, c   and with the following properties: (i) The origin is the unique equilibrium for all parameters. (ii) Ifitshape a crosses zero, then the origin changes its stability, and a limit cycle bifurcates from the euqilibrium. We inverstigate analytically this bifurcation in dependence on the parameters itshape b and itshape c and establish the existence of families of limit cycles of multiplicity one, two and three bifurcating from the origin.

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