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In this paper we prove that families of germs of one-parameter analytic differential equations with a Hopf bifurcation are rigid in the parameter. These systems are intrinsically real: the complex phase portrait is invariant under an antiholomorphic involution. The latter permits to identify the modulus of the analytic classification. The dynamics in the Siegel domain does not explain the existence of the real structure. Rather, these properties are meaningful in the Poincaré domain, where the fixed points are linearizable.
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Waldo Arriagada, João Fialho, Parametric rigidness of germs of analytic unfoldings with a Hopf bifurcation. Port. Math. 73 (2016), no. 2, pp. 153–170DOI 10.4171/PM/1981