Superstable periodic  orbits of 1d maps under quasi-periodic forcing and reducibility  loss

Àngel Jorba; Pau Rabassa; Joan Carles Tatjer

doi:10.3934/dcds.2014.34.589

Article Contents

2014, Volume 34, Issue 2: 589-597. Doi: 10.3934/dcds.2014.34.589

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Superstable periodic orbits of 1d maps under quasi-periodic forcing and reducibility loss

1.
Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585 , 08007 Barcelona
2.
Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, 08007 Barcelona,
3.
Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via, 585, 08080 Barcelona

Received: January 31, 2013

Revised: March 31, 2013

Published: January 2014

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Abstract

Let $g_{\alpha}$ be a one-parameter family of one-dimensional maps with a cascade of period doubling bifurcations. Between each of these bifurcations, a superstable periodic orbit is known to exist. An example of such a family is the well-known logistic map. In this paper we deal with the effect of a quasi-periodic perturbation (with only one frequency) on this cascade. Let us call $\varepsilon$ the perturbing parameter. It is known that, if $\varepsilon$ is small enough, the superstable periodic orbits of the unperturbed map become attracting invariant curves (depending on $\alpha$ and $\varepsilon$) of the perturbed system. In this article we focus on the reducibility of these invariant curves.
The paper shows that, under generic conditions, there are both reducible and non-reducible invariant curves depending on the values of $\alpha$ and $\varepsilon$. The curves in the space $(\alpha,\varepsilon)$ separating the reducible (or the non-reducible) regions are called reducibility loss bifurcation curves. If the map satifies an extra condition (condition satisfied by the quasi-periodically forced logistic map) then we show that, from each superattracting point of the unperturbed map, two reducibility loss bifurcation curves are born. This means that these curves are present for all the cascade.

Keywords:

Invariant curves,
bifurcation cascades.

Mathematics Subject Classification: Primary: 37C55; Secondary: 37E99, 37G35.

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References

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