A renormalization operator for 1D maps under quasi-periodic perturbations. (April 2015) - Jorba, À; Rabassa, P; Tatjer, J C
Abstract This paper concerns the reducibility loss of (periodic) invariant curves of quasi-periodically forced one-dimensional maps and its relationship with the renormalization operator. Let gα be a one-parametric family of one-dimensional maps with a cascade of period doubling bifurcations. Between each of these bifurcations, there exists a parameter value αn such that <?CDATA $g_{α_n}$ ?> <inline-graphic xlink:href="ark:/27927/pgj2cb1tt5t" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> has a superstable periodic orbit of period 2n. Consider a quasi-periodic perturbation (with only one frequency) of the one-dimensional family of maps, and let us call ε the perturbing parameter. For ε small enough, the superstable periodic orbits of the unperturbed map become attracting invariant curves (depending on α and ε) of the perturbed system. Under a suitable hypothesis, it is known that there exist two reducibility loss bifurcation curves around each parameter value (αn, 0), which can be locally expressed as <?CDATA $(α_n^+(ε), ε)$ ?> <inline-graphic xlink:href="ark:/27927/pgj2cb1v2vz" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> and <?CDATA $(α_n^-(ε), ε)$ ?> <inline-graphic xlink:href="ark:/27927/pgj2cb1v4m4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />. We propose an extension of the classic one-dimensional (doubling) renormalization operator to the quasi-periodic case. We show that this extension is well defined and the operator is differentiable. Moreover, we show that the slopes of reducibility loss bifurcation <?CDATA $\frac{\rmd}{\rmdε} α_n^±(0)$ ?> <inline-graphic xlink:href="ark:/27927/pgj2cb1trgr" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> can be written in terms of the tangent map of the new quasi-periodic renormalization operator. In particular, our result applies to the families of quasi-periodic forced perturbations of the Logistic Map typically encountered in the literature. We also present a numerical study that demonstrates that the asymptotic behaviour of <?CDATA $\{\frac{\rmd}{\rmdε} α_n^±(0)\}_{n≥ 0}$ ?> <inline-graphic xlink:href="ark:/27927/pgj2cb1trsq" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> is governed by the dynamics of the proposed quasi-periodic renormalization operator.