An extensive search for stable periodic orbits (sinks) for the Hénon map in a small neighborhood of the classical parameter values is carried out. Several parameter values which generate a sink are found and verified by rigorous numerical computations. Each found parameter value is extended to a larger region of existence using a simplex continuation method. The structure of these regions of existence is investigated. This study shows that for the Hénon map, there exist sinks close to the classical case.
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