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

Jaume Llibre; Amar Makhlouf; Sabrina Badi

doi:10.3934/dcds.2009.25.1287

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2009, Volume 25, Issue 4: 1287-1295. Doi: 10.3934/dcds.2009.25.1287

This issue Previous Article Strong traces for degenerate parabolic-hyperbolic equations Next Article Reflection of highly oscillatory waves with continuous oscillatory spectra for semilinear hyperbolic systems

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

1.
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona
2.
Department of Mathematics, University of Annaba, P.O.Box 12, Annaba 23000
3.
Department of Mathematics, University of Guelma, P.O.Box 401, Guelma 24000

Received: October 31, 2008

Revised: May 31, 2009

Published: October 2009

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Abstract

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.

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Mathematics Subject Classification: Primary: 37G15; Secondary: 37D45.

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