Given a quadratic polynomial P with a fixed Siegel disk D, given a P− invariant subdisk D0, and a scale , we provide special perturbations of the parameter such that the new quadratic polynomial P 0 has a Siegel disk D0 with the following properties: - The rotation number of D0 is close to that of D - D0 contains D0 - P0 has a cycle Hausdorff close to the boudary of D0 - D0 roughly covers more than half the area of all balls of radius contained in D If the rotation number of D has all its continued fraction entries greater than M, then the same holds for our perturbations D0.