A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Numerical algorithms
Àlex Haro Rafael de la Llave
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.
keywords: Quasi-periodic systems numerical methods. invariant manifolds invariant tori
A note on the fractalization of saddle invariant curves in quasiperiodic systems
Jordi-Lluís Figueras Àlex Haro
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.
keywords: Breakdown of saddle invariant curves quasiperiodic systems.
On strange attractors in a class of pinched skew products
Àlex Haro
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.
keywords: pinched skew products. Strange attractors strange non-chaotic attractors
Triple collisions of invariant bundles
Jordi-Lluís Figueras Àlex Haro
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.
keywords: Oseledets theory. Nonuniformly hyperbolic linear skew-products
One dimensional invariant manifolds of Gevrey type in real-analytic maps
I. Baldomá Àlex Haro
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 $C^\infty$.
keywords: asymptotic series Gevrey. parabolic points Invariant manifolds

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