Hopf bifurcation in higher dimensional differential systems via the averaging method

• Autores: Jaume Llibre , Xiang Zhang
• Localización: Pacific journal of mathematics, ISSN 0030-8730, Vol. 240, Nº 2, 2009, págs. 321-341
• Idioma: inglés
• DOI: 10.2140/pjm.2009.240.321
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• Resumen

  • We study the Hopf bifurcation of C3 differential systems in Rn showing that l limit cycles can bifurcate from one singularity with eigenvalues ±bi and n - 2 zeros with l in {0,1,�,2n-3}. As far as we know this is the first time that it is proved that the number of limit cycles that can bifurcate in a Hopf bifurcation increases exponentially with the dimension of the space. To prove this result, we use first-order averaging theory. Further, in dimension 4 we characterize the shape and the kind of stability of the bifurcated limit cycles. We apply our results to certain fourth-order differential equations and then to a simplified Marchuk model that describes immune response.