Universal Scalings in Homoclinic Doubling Cascades

Cascades of period doubling bifurcations are found in one parameter families of differential equations in ℝ3. When varying a second parameter, the periodic orbits in the period doubling cascade can disappear in homoclinic bifurcations. In one of the possible scenarios one finds cascades of homoclinic doubling bifurcations. Relevant aspects of this scenario can be understood from a study of interval maps close to x↦p+r(1 −xβ)2, β∈ (½,1). We study a renormalization operator for such maps. For values of β close to ½, we prove the existence of a fixed point of the renormalization operator, whose linearization at the fixed point has two unstable eigenvalues. This is in marked contrast to renormalization theory for period doubling cascades, where one unstable eigenvalue appears. From the renormalization theory we derive consequences for universal scalings in the bifurcation diagrams in the parameter plane.