Joan Carles Tatjer

OLD AND NEW RESULTS ON STRANGE NONCHAOTIC ATTRACTORS

Classical and new results concerning the topological structure of skew-products semiflows, coming from nonautonomous maps and differential equations, are combined in order to establish rigorous conditions giving rise to the occurrence of strange nonchaotic attractors on 𝕋d × ℝ. A special attention is paid to the relation of these sets with the almost automorphic extensions of the base flow. The scope of the results is clarified by applying them to the Harper map, although they are valid in a much wider context.

Birth of discrete Lorenz attractors at the bifurcations of 3D maps with homoclinic tangencies to saddle points

It was established in [1] that bifurcations of three-dimensional diffeomorphisms with a homoclinic tangency to a saddle-focus fixed point with the Jacobian equal to 1 can lead to Lorenz-like strange attractors. In the present paper we prove an analogous result for three-dimensional diffeomorphisms with a homoclinic tangency to a saddle fixed point with the Jacobian equal to 1, provided the quadratic homoclinic tangency under consideration is nonsimple.It was established in [1] that bifurcations of three-dimensional diffeomorphisms with a homoclinic tangency to a saddle-focus fixed point with the Jacobian equal to 1 can lead to Lorenz-like strange attractors. In the present paper we prove an analogous result for three-dimensional diffeomorphisms with a homoclinic tangency to a saddle fixed point with the Jacobian equal to 1, provided the quadratic homoclinic tangency under consideration is non-simple.

Chaotic dynamics for two-dimensional tent maps

For a two-dimensional extension of the classical one-dimensional family of tent maps, we prove the existence of an open set of parameters for which the respective transformation presents a strange attractor with two positive Lyapounov exponents. Moreover, periodic orbits are dense on this attractor and the attractor supports a unique ergodic invariant probability measure.

Piecewise Linear Bidimensional Maps as Models of Return Maps for 3D Diffeomorphisms

The goal of this paper is to study the dynamics of a simple family of piecewise linear maps in dimension two, that we call Expanding Baker Maps (EBM), which is a simplified model of a quadratic limit return map which appears in the study of certain homoclinic bifurcations of two-parameter families of three-dimensional dissipative diffeomorphisms. In spite of its simplicity the EBM capture some of the more relevant dynamics of the quadratic family, specially that related to the evolution of 2D strange attractors.

A NUMERICAL STUDY OF UNIVERSALITY AND SELF-SIMILARITY IN SOME FAMILIES OF FORCED LOGISTIC MAPS

We explore different two-parametric families of quasi-periodically Forced Logistic Maps looking for universality and self-similarity properties. In the bifurcation diagram of the one-dimensional Logistic Map, it is well known that there exist parameter values sn where the 2n-periodic orbit is superattracting. Moreover, these parameter values lay between the parameters corresponding to two consecutive period doublings. In the quasi-periodically Forced Logistic Maps, these points are replaced by invariant curves, that undergo a (finite) sequence of period doublings. In this work, we study numerically the presence of self-similarities in the bifurcation diagram of the invariant curves of these quasi-periodically Forced Logistic Maps. Our computations show a remarkable self-similarity for some of these families. We also show that this self-similarity cannot be extended to any quasi-periodic perturbation of the Logistic map.

On non-smooth pitchfork bifurcations in invertible quasi-periodically forced 1-D maps

In this note we revisit an example introduced by T. Jäger in which a Strange Non-chaotic Attractor seems to appear during a pitchfork bifurcation of invariant curves in a quasi-periodically forced 1-d map. In this example, it is remarkable that the map is invertible and, hence, the invariant curves are always reducible. In the first part of the paper we give a numerical description (based on a precise computation of invariant curves and Lyapunov exponents) of the phenomenon. The second part consists in a preliminary study of the phenomenon, in which we prove that an analytic self-symmetric invariant curve is persistent under perturbations.

A renormalization operator for 1D maps under quasi-periodic perturbations

This paper concerns the reducibility loss of (periodic) invariant curves of quasi-periodically forced one-dimensional maps and its relationship with the renormalization operator. Let gα be a one-parametric family of one-dimensional maps with a cascade of period doubling bifurcations. Between each of these bifurcations, there exists a parameter value αn such that has a superstable periodic orbit of period 2n. Consider a quasi-periodic perturbation (with only one frequency) of the one-dimensional family of maps, and let us call e the perturbing parameter. For e small enough, the superstable periodic orbits of the unperturbed map become attracting invariant curves (depending on α and e) of the perturbed system. Under a suitable hypothesis, it is known that there exist two reducibility loss bifurcation curves around each parameter value (αn, 0), which can be locally expressed as and . We propose an extension of the classic one-dimensional (doubling) renormalization operator to the quasi-periodic case. We show that this extension is well defined and the operator is differentiable. Moreover, we show that the slopes of reducibility loss bifurcation can be written in terms of the tangent map of the new quasi-periodic renormalization operator. In particular, our result applies to the families of quasi-periodic forced perturbations of the Logistic Map typically encountered in the literature. We also present a numerical study that demonstrates that the asymptotic behaviour of is governed by the dynamics of the proposed quasi-periodic renormalization operator.

On the divergence of polynomial interpolation

Consider a triangular interpolation scheme on a continuous piecewise C1 curve of the complex plane, and let Γ be the closure of this triangular scheme. Given a meromorphic function f with no singularities on Γ, we are interested in the region of convergence of the sequence of interpolating polynomials to the function f. In particular, we focus on the case in which Γ is not fully contained in the interior of the region of convergence defined by the standard logarithmic potential. Let us call Γout the subset of Γ outside of the convergence region.In the paper we show that the sequence of interpolating polynomials, {Pn}n, is divergent on all the points of Γout, except on a set of zero Lebesgue measure. Moreover, the structure of the set of divergence is also discussed: the subset of values z for which there exists a partial sequence of {Pn(z)}n that converges to f(z) has zero Hausdorff dimension (so it also has zero Lebesgue measure), while the subset of values for which all the partials are divergent has full Lebesgue measure.The classical Runge example is also considered. In this case we show that, for all z in the part of the interval (-5,5) outside the region of convergence, the sequence {Pn(z)}n is divergent.