Abstract We study a situation close to the Sil'nikov-Hopf bifurcation, placing special emphasis on the attractors. A model is proposed for the return map, to a suitable surface, which explains some numerical results concerning a particular physical system. The model is quite general and can give reasonable explanations to many other examples. Later on we introduce a circle map as an additional simplification if the return map is strongly dissipative near the attractor. This circle map has very interesting properties: It plays a role similar to the logistic map in the case of maps on the interval. However it doesn't have the disadvantage of escapes to infinity produced in the logistic map when the parameter is large enough.
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