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### Open Access Journals

DCDS-B

For certain 3D-homoclinic tangencies where the unstable manifold of the saddle point involved in the homoclinic tangency has dimension two, many different strange attractors have been numerically observed for the corresponding family of limit return maps. Moreover, for some special value of the parameter, the respective limit return map is conjugate to what was called bidimensional tent map. This piecewise affine map is an example of what we call now Expanding Baker Map, and the main objective of this paper is to show how many of the different attractors exhibited for the limit return maps resemble the ones observed for Expanding Baker Maps.

DCDS-B

We introduce a scenario for the fractalization of invariant curves for a special class of quasi-periodically forced 1-D maps. In this situation, a smooth invariant curve becomes increasingly wrinkled when its Lyapunov exponent goes to zero, but it keeps being smooth as long as its exponent is negative. It is remarkable that the curve becomes so wrinkled that numerical simulations may not distinguish the curve from a strange attracting set.

Moreover, we show that a nonreducible invariant curve with a positive Lyapunov exponent is not persistent in a general quasi-periodically forced 1-D map. We also derive some new results on the behaviour of the Lyapunov exponent of an invariant curve w.r.t. parameters.

The paper contains some numerical examples. One of them is based on the quasi-periodically forced logistic map, where we show numerically that the fractalization of an invariant curve of this system may fit into our scenario.

Moreover, we show that a nonreducible invariant curve with a positive Lyapunov exponent is not persistent in a general quasi-periodically forced 1-D map. We also derive some new results on the behaviour of the Lyapunov exponent of an invariant curve w.r.t. parameters.

The paper contains some numerical examples. One of them is based on the quasi-periodically forced logistic map, where we show numerically that the fractalization of an invariant curve of this system may fit into our scenario.

DCDS-B

We numerically analyse different kinds of one-dimensional and two-dimensional attractors for the limit return map associated to the unfolding of homoclinic tangencies for a large class of three-dimensional dissipative diffeomorphisms. Besides describing the topological properties of these attractors, we often numerically compute their Lyapunov exponents in order to clarify where two-dimensional strange attractors can show up in the parameter space. Hence, we are specially interested in the case in which the unstable manifold of the periodic saddle taking part in the homoclinic tangency has dimension two.

DCDS-B

We study the dynamics of the Forced Logistic Map in the cylinder. We
compute a bifurcation diagram in terms of the dynamics of the
attracting set. Different properties of the attracting set are
considered, such as the Lyapunov exponent and, in the case of having a
periodic invariant curve, its period and reducibility. This
reveals that the parameter values for which the invariant curve
doubles its period are contained in regions of the parameter space
where the invariant curve is reducible. Then we present two additional
studies to explain this fact. In first place we consider the images
and the preimages of the critical set (the set where the derivative of
the map w.r.t the non-periodic coordinate is equal to zero). Studying
these sets we construct constrains in the parameter space for the
reducibility of the invariant curve. In second place we consider the
reducibility loss of the invariant curve as a codimension one
bifurcation and we study its interaction with the period doubling
bifurcation. This reveals that, if the reducibility loss and the
period doubling bifurcation curves meet, they do it in a tangent way.

keywords:
fractalization
,
bifurcation cascades
,
skew products
,
Invariant curves
,
reducibility loss.

DCDS

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.

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.

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