Period doubling and reducibility in the quasi-periodically forced logistic map

Angel Jorba; Pau Rabassa; Joan Carles Tatjer

Home/Archive/arXiv

arXiv · 1112.4065

Period doubling and reducibility in the quasi-periodically forced logistic map

Angel Jorba·Pau Rabassa·Joan Carles Tatjer

View original source ↗Read source document ↗Export metadata ↓

Abstract

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, as the Lyapunov exponent and, in the case of having a periodic invariant curve, its period and its 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 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.

Explore related subjects

math.DS

Keep this discovery

BibTeX RIS

Angel Jorba, Pau Rabassa, Joan Carles Tatjer. 2011-12-17. Period doubling and reducibility in the quasi-periodically forced logistic map. https://arxiv.org/abs/1112.4065

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Admissible Fourier Lengths, KAM Reducibility, and Spectral Applications

We develop a perturbative KAM reducibility theory for one-frequency $\mathrm{SL}(2,\mathbb{R})$ cocycles based on an admissible Fourier length $\ell$. The regularity relevant to the iteration is measured by positive adapted Fourier width rather than ordinary smoothness in the Euclidean length $|n|$. The same length governs Fourier decay, truncation and resonance scales, and the arithmetic condition controlling the small divisors. This framework contains the classical analytic and Gevrey settings, while non-monotone choices of $\ell$ allow classical nowhere differentiable Weierstrass-type perturbations and continuous perturbations outside every positive H\"older class. As spectral applications, we obtain purely absolutely continuous spectrum for every phase and $1/2$-H\"older continuity of the integrated density of states for the associated quasiperiodic Schr\"odinger operators. The Aubry dual has pure point spectrum for Lebesgue almost every dual phase, with eigenfunctions exponentially localized in the metric induced by $\ell$. We also construct nowhere differentiable quasiperiodic potentials with purely absolutely continuous Cantor spectrum.

math.DS↗

arXiv2026

Dynamics inside the attracting basins of some skew products

Polynomial skew products in $\mathbb{C}^2$ are maps of the form $F(z,w)=(P(z),Q(z,w))$, where $P$ and $Q$ are polynomials. Their local dynamics have been widely investigated. In this paper, we study the global dynamics inside Fatou components of some skew products. We consider all the inverse images in a Fatou component of a given point and use the Kobayashi metric to measure the distance between points. In the cases we consider, there are always arbitrarily large Kobayashi balls in the complement of these inverse sets.

math.DS↗

arXiv2026

Ergodicity of dynamical systems without uniqueness of orbits

Recently, there has been considerable interest in the study of non-deterministic dynamical systems. To analyze the chaotic behavior of such systems from a measure-theoretic viewpoint, it is desirable to consider ergodicity. However, the classical definition of ergodicity involves invariant sets, whose definition is not unique for non-deterministic dynamical systems. Thus, we are led to the question of which invariance yields an interesting definition of ergodicity. Here, we propose a definition based on the strong backward invariance and show that analogs of classical results hold. We also consider implications of the Birkhoff ergodic theorem for systems without uniqueness of orbits.

math.DS↗