Universal scalings in homoclinic doubling cascades

We study a renormalization operator for families of one dimensional maps close to x 7! p + r(1 x ) 2 , 2 ( 1 2 ; 1). Such functions occur in the study of cascades of homoclinic doubling bifurcations in three dimensional di erential equations. For values of close to 1 2 , we prove the existence of a xed point of the renormalization operator, whose linearization at the xed point has two unstable eigenvalues. This is in marked contrast to renormalization theory for period doubling cascades, in which one unstable eigenvalue appears. We derive from the renormalization theory consequences for universal scalings in the bifurcation diagrams.