A note on the fractalization of saddle invariant curves in quasiperiodic systems

Dynamical systems appear in many models in all sciences and in technology. They can be either discrete or continuous, finite or infinite dimensional, deterministic or with random terms. Many theoretical results, the related algorithms and implementations for careful simulations and a wide range of applications have been obtained up to now. But still many key questions remain open. They are mainly related either to global aspects of the dynamics or to the lack of a sufficiently good agreement between qualitative and quantitative results. In these notes a sample of questions, for which the author is not aware of the existence of a good solution, are presented. Of course, it is easy to largely extend the list.

“…These can be detected by looking at several observables, like the angles between bundles, the Lyapunov multipliers or the blow up of the Sobolev norm. See [19,34,35,57,58,68].…”

mentioning

confidence: 99%

Some questions looking for answers in dynamical systems

Simó¹

2018

“…Here we use Definition 5.1 because it gives a straightforward interpretation of the numerical computations shown in Figure 2. Nevertheless, numerical evidence suggests that the examples in this paper also fractalize according to [4].…”

mentioning

confidence: 74%

“…The second map is uniquely ergodic if ω 2π is irrational, with the Lebesgue measure as the unique invariant measure and the first map is uniquely ergodic if ω, ρ and 2π are rationally independent. 4. Normal forms and Lyapunov exponents.…”

Section: If We Write (R/z)mentioning

confidence: 99%

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Classification of linear skew-products of the complex plane and an affine route to fractalization

Fagella¹,

Jorba²,

Jorba-Cuscó³

et al. 2019

Linear skew products of the complex plane, θ → θ + ω, z → a(θ)z, where θ ∈ T, z ∈ C, ω 2π is irrational, and θ → a(θ) ∈ C \ {0} is a smooth map, appear naturally when linearizing dynamics around an invariant curve of a quasi-periodically forced complex map. In this paper we study linear and topological equivalence classes of such maps through conjugacies which preserve the skewed structure, relating them to the Lyapunov exponent and the winding number of θ → a(θ). We analyze the transition between these classes by considering one parameter families of linear skew products. Finally, we show that, under suitable conditions, an affine variation of the maps above has a non-reducible invariant curve that undergoes a fractalization process when the parameter goes to a critical value. This phenomenon of fractalization of invariant curves is known to happen in nonlinear skew products, but it is remarkable that it also occurs in simple systems as the ones we present.

“…The linear asymptotics for the distance in assumption A2b enters in the last step, and also affect the final asymptotics. As we said in (5), the minimum angle between the invariant directions is conjectured to be linear when a system loses uniform hyperbolicity (see for instance [7]).…”

Section: Discussion and Future Directionsmentioning

confidence: 94%

Sharp <inline-formula><tex-math id="M1">\begin{document}$ \frac12 $\end{document}</tex-math></inline-formula>-Hölder continuity of the Lyapunov exponent at the bottom of the spectrum for a class of Schrödinger cocycles

Figueras¹,

Timoudas²

2020

Self Cite

We consider a similar type of scenario for the disappearance of uniform of hyperbolicity as in Bjerklöv and Saprykina (2008 Nonlinearity 21), where it was proved that the minimum distance between invariant stable and unstable bundles has a linear power law dependence on parameters. In this scenario we prove that the Lyapunov exponent is sharp 1 2 -Hölder continuous. In particular, we show that the Lyapunov exponent of Schrödinger cocycles with a potential having a unique non-degenerate minimum, is sharp 1 2 -Hölder continuous below the lowest energy of the spectrum, in the large coupling regime.