A numerical Study of Universality and Self-Similarity in some Families of forced logistic Maps

We explore different two-parametric families of quasi-periodically Forced Logistic Maps looking for universality and self-similarity properties. In the bifurcation diagram of the one-dimensional Logistic Map, it is well known that there exist parameter values sn where the 2n-periodic orbit is superattracting. Moreover, these parameter values lay between the parameters corresponding to two consecutive period doublings. In the quasi-periodically Forced Logistic Maps, these points are replaced by invariant curves, that undergo a (finite) sequence of period doublings. In this work, we study numerically the presence of self-similarities in the bifurcation diagram of the invariant curves of these quasi-periodically Forced Logistic Maps. Our computations show a remarkable self-similarity for some of these families. We also show that this self-similarity cannot be extended to any quasi-periodic perturbation of the Logistic map.