Non-smooth saddle-node bifurcations II: Dimensions of strange attractors

Abstract

We study the geometric and topological properties of strange non-chaotic attractors created in non-smooth saddle-node bifurcations of quasiperiodically forced interval maps. By interpreting the attractors as limit objects of the iterates of a continuous curve and controlling the geometry of the latter, we determine their Hausdorff and box-counting dimension and show that these take distinct values. Moreover, the same approach allows to describe the topological structure of the attractors and to prove their minimality.

Rectifiability of a class of invariant measures with one non-vanishing Lyapunov exponent

“…With this article, we show that in many cases the measures corresponding to strange non-chaotic attractors are in fact one-rectifiable, that is, they are absolutely continuous with respect to the restriction of the one-dimensional Hausdorff measure to a countable union of Lipschitz graphs (see Section 2.1 for the exact definition). Similar results have already been obtained in previous studies [5,6]. However, the underlying geometric picture of the proof in the present case differs to a large extent and makes the authors believe that rectifiability should be expected also in more general situations.…”

“…In this setting, Young [16] and Bjerklöv [17,18] developed powerful methods-in the spirit of the multiscale analysis and parameter exclusion techniques by Benedicks and Carleson [19]-to examine the occurrence and properties of SNA's. These methods had later been adapted to non-linear systems (such as ( * )) in [20,21,22,6].…”

“…Proposition 3.1). On a combinatorial level, the strategy we pursue has been applied successfully to ergodic measures supported on SNA/SNR-pairs that occur in so-called saddle-node bifurcations of qpf monotone interval maps [6]. These are skew-products similar to ( * ) but defined on T 1 × [0, 1] and such that the maps f θ (·) are monotonously increasing (for each fixed θ ∈ T 1 ).…”

“…These are skew-products similar to ( * ) but defined on T 1 × [0, 1] and such that the maps f θ (·) are monotonously increasing (for each fixed θ ∈ T 1 ). We will thus be able to recycle the rather technical combinatorial findings of [6].…”

Effects of Periodic Forcing on a Paleoclimate Delay Model

“…We therefore conjecture that the transition is an effect of the near-quasiperiodic modulation of the basic periodic forcing studied in this paper, which is present in the astronomical forcing, resulting in a more complicated version of the rate-induced tipping phenomenon obtained for the step-wise amplitude modulation (12). This may require application of the general quasiperiodic theory of Fuhrmann et al [14] or the pullback attractor framework outlined in Chekroun et al [7]. This appendix gives a brief explanation for the extension of the phase space of DDEs, from which we permit initial conditions of the stroboscopic map M in subsections 3.2 to 3.4.…”