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### Open Access Journals

DCDS-B

In this paper we develop several numerical algorithms for the
computation of invariant manifolds in quasi-periodically forced
systems. The invariant manifolds we consider are invariant tori and
the asymptotic invariant manifolds (whiskers) to these tori.

The algorithms are based on the parameterization method described in [36], where some rigorous results are proved. In this paper, we concentrate on numerical issues of algorithms. Examples of implementations appear in the companion paper [34].

The algorithms for invariant tori are based essentially on Newton method, but taking advantage of dynamical properties of the torus, such as hyperbolicity or reducibility as well as geometric properties.

The algorithms for whiskers are based on power-matching expansions of the parameterizations. Whiskers include as particular cases the usual (strong) stable and (strong) unstable manifolds, and also, in some cases, the slow manifolds which dominate the asymptotic behavior of solutions converging to the torus.

The algorithms are based on the parameterization method described in [36], where some rigorous results are proved. In this paper, we concentrate on numerical issues of algorithms. Examples of implementations appear in the companion paper [34].

The algorithms for invariant tori are based essentially on Newton method, but taking advantage of dynamical properties of the torus, such as hyperbolicity or reducibility as well as geometric properties.

The algorithms for whiskers are based on power-matching expansions of the parameterizations. Whiskers include as particular cases the usual (strong) stable and (strong) unstable manifolds, and also, in some cases, the slow manifolds which dominate the asymptotic behavior of solutions converging to the torus.

DCDS-S

The purpose of this paper is to describe a new mechanism of destruction of
saddle invariant curves in quasiperiodically forced systems, in which an
invariant curve experiments a process of fractalization, that is, the curve
gets increasingly wrinkled until it breaks down. The phenomenon resembles the
one described for attracting invariant curves in a number of quasiperiodically
forced dissipative systems, and that has received the attention in the
literature for its connections with the so-called Strange Non-Chaotic
Attractors. We present a general conceptual framework that provides a simple
unifying mathematical picture for fractalization routes in dissipative and
conservative systems.

DCDS

In this paper we construct strange attractors in a class of
pinched skew product dynamical systems over homeomorphims
on a compact metric space. We assume that
maps between fibers satisfy Inada conditions and that the base space
is a super-repeller (it is invariant and its vertical Lyapunov exponent is $+\infty$).
In particular, we prove the existence
of a measurable but non-continuous invariant graph, whose vertical
Lyapunov exponent
is negative. %We will refer to such an object as a strange attractor.

Since the dynamics on the strange attractor is the one given by the base homeomorphism, we will say that it is a strange chaotic attractor or a strange non-chaotic attractor depending on the fact that the dynamics on the base is chaotic or non-chaotic. The results complement the paper by G. Keller on rigorous proofs of existence of strange non-chaotic attractors.

Since the dynamics on the strange attractor is the one given by the base homeomorphism, we will say that it is a strange chaotic attractor or a strange non-chaotic attractor depending on the fact that the dynamics on the base is chaotic or non-chaotic. The results complement the paper by G. Keller on rigorous proofs of existence of strange non-chaotic attractors.

DCDS-B

We provide several explicit examples of 3D quasiperiodic
linear skew-products with simple Lyapunov spectrum, that is with
$3$ different Lyapunov multipliers, for which the corresponding Oseledets bundles
are measurable but not continuous, colliding in a measure zero dense set.

DCDS-B

In this paper we study the basic questions of existence, uniqueness,
differentiability, analyticity and computability of one dimensional
parabolic manifolds of degenerate fixed points, i.e. invariant
manifolds tangent to the eigenspace of 1, which is assumed to be a
simple eigenvalue. We use the parameterization method, reducing the
dynamics on the parabolic manifold to a polynomial. We prove that
the asymptotic expansions of the parabolic manifold are of Gevrey
type. Moreover, under suitable hypothesis, we also prove that the
asymptotic expansions correspond to a real-analytic parameterization
of an invariant curve that goes to the fixed point. The
parameterization is Gevrey at the fixed point, hence $C^\infty$.

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