Resumen de Hopf bifurcation in higher dimensional differential systems via the averaging method

Jaume Llibre , Xiang Zhang

• 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.