A renormalization operator for 1D maps under quasi-periodic perturbations. (April 2015)

• Record Type: Journal Article
• Authors: Jorba, À; Rabassa, P; Tatjer, J C
• Is Part Of: Nonlinearity. Volume 28:Number 4(2015:Apr.)
• Page Start: 1017
• Page End: 1042
• Publication Date: 2015-04
• Subjects: Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/0951-7715/28/4/1017
• Languages: English
• ISSNs: 0951-7715

Abstract

<abstract> <title>Abstract</title> <p>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 <italic>g</italic><sub><italic>α</italic></sub> 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 <italic>α</italic><sub><italic>n</italic></sub> such that <inline-formula><tex-math><?CDATA $g_{\alpha_n}$ ?></tex-math><inline-graphic xlink:href="ark:/27927/pgj2cb1tt5t" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /></inline-formula> has a superstable periodic orbit of period 2<sup><italic>n</italic></sup>. Consider a quasi-periodic perturbation (with only one frequency) of the one-dimensional family of maps, and let us call <italic>ε</italic> the perturbing parameter. For <italic>ε</italic> small enough, the superstable periodic orbits of the unperturbed map become attracting invariant curves (depending on <italic>α</italic> and <italic>ε</italic>) of the perturbed system. Under a suitable hypothesis, it is known that there exist two reducibility loss bifurcation curves around each parameter value (<italic>α</italic><sub><italic>n</italic></sub>, 0), which can be locally expressed as <inline-formula><tex-math><?CDATA $(\alpha_n^+(\varepsilon), \varepsilon)$ ?></tex-math><inline-graphic xlink:href="ark:/27927/pgj2cb1v2vz" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /></inline-formula> and <inline-formula><tex-math><?CDATA $(\alpha_n^-(\varepsilon), \varepsilon)$ ?></tex-math><inline-graphic xlink:href="ark:/27927/pgj2cb1v4m4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /></inline-formula>. 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 <inline-formula><tex-math><?CDATA $\frac{\rmd}{\rmd\varepsilon} \alpha_n^\pm(0)$ ?></tex-math><inline-graphic xlink:href="ark:/27927/pgj2cb1trgr" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /></inline-formula> 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 <inline-formula><tex-math><?CDATA $\{\frac{\rmd}{\rmd\varepsilon} \alpha_n^\pm(0)\}_{n\geq 0}$ ?></tex-math><inline-graphic xlink:href="ark:/27927/pgj2cb1trsq" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /></inline-formula> is governed by the dynamics of the proposed quasi-periodic renormalization operator.</p> </abstract>