TITLE:
A numerical study of universality and self-similarity in the forced
logistic map

AUTHORS:
Pau Rabassa^1, Angel Jorba^2, Joan Carles Tatjer^2

(1) Johann Bernoulli Institute for Mathematics and Computer Science
    University of Groningen
    PO Box 407
    9700 AK Groningen, The Netherlands.

(2) Departament de Matematica Aplicada i Analisi
    Universitat de Barcelona
    Gran Via 585
    08007 Barcelona, Spain.

E-mails: p.rabassa.sans@rug.nl, angel@maia.ub.es, jcarles@maia.ub.es

ABSTRACT:
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 $s_n$ where the
$2^n$-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.