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| 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. We will
focus on the effect of a quasi-periodic perturbation on this cascade.
It is known that, if $\varepsilon$ is small enough, the superstable
periodic orbits of the unperturbed map become attracting invariant
curves the perturbed system. This talk will focus on the reducibility
of these invariant curves.
We will show 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 and the non-reducible
regions are called reducibility loss bifurcation curves.
We will discuss on the existence of this bifurcation curves along the
cascade of the one-dimensional map and its asymptotic behavior. |
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