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December  2009, 25(4): 1287-1295. doi: 10.3934/dcds.2009.25.1287

$3$ - dimensional Hopf bifurcation via averaging theory of second order

Jaume Llibre 1, , Amar Makhlouf 2, and  Sabrina Badi 3,

1. 

Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona
2. 

Department of Mathematics, University of Annaba, P.O.Box 12, Annaba 23000, Algeria
3. 

Department of Mathematics, University of Guelma, P.O.Box 401, Guelma 24000, Algeria

Received  November 2008 Revised  June 2009 Published  September 2009

We study the Hopf bifurcation occurring in polynomial quadratic vector fields in $\R^3$. By applying the averaging theory of second order to these systems we show that at most $3$ limit cycles can bifurcate from a singular point having eigenvalues of the form $\pm bi$ and $0$. We provide an example of a quadratic polynomial differential system for which exactly $3$ limit cycles bifurcate from a such singular point.
Keywords: Limit cycle, Averaging theory., Hopf bifurcation.
Mathematics Subject Classification: Primary: 37G15; Secondary: 37D4.
Citation: Jaume Llibre, Amar Makhlouf, Sabrina Badi. $3$ - dimensional Hopf bifurcation via averaging theory of second order. Discrete & Continuous Dynamical Systems, 2009, 25 (4) : 1287-1295. doi: 10.3934/dcds.2009.25.1287
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