Density of mode-locking property for quasi-periodically forced Arnold circle maps

Pub Date : 2024-04-04 DOI:10.1017/etds.2024.27

JIAN WANG, ZHIYUAN ZHANG

{"title":"Density of mode-locking property for quasi-periodically forced Arnold circle maps","authors":"JIAN WANG, ZHIYUAN ZHANG","doi":"10.1017/etds.2024.27","DOIUrl":null,"url":null,"abstract":"We show that the mode-locking region of the family of quasi-periodically forced Arnold circle maps with a topologically generic forcing function is dense. This gives a rigorous verification of certain numerical observations in [M. Ding, C. Grebogi and E. Ott. Evolution of attractors in quasiperiodically forced systems: from quasiperiodic to strange nonchaotic to chaotic. Phys. Rev. A 39(5) (1989), 2593–2598] for such forcing functions. More generally, under some general conditions on the base map, we show the density of the mode-locking property among dynamically forced maps (defined in [Z. Zhang. On topological genericity of the mode-locking phenomenon. Math. Ann. 376 (2020), 707–72]) equipped with a topology that is much stronger than the $C^0$ topology, compatible with smooth fiber maps. For quasi-periodic base maps, our result generalizes the main results in [A. Avila, J. Bochi and D. Damanik. Cantor spectrum for Schrödinger operators with potentials arising from generalized skew-shifts. Duke Math. J.146 (2009), 253–280], [J. Wang, Q. Zhou and T. Jäger. Genericity of mode-locking for quasiperiodically forced circle maps. Adv. Math.348 (2019), 353–377] and Zhang (2020).","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/etds.2024.27","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

引用次数: 0

Abstract

We show that the mode-locking region of the family of quasi-periodically forced Arnold circle maps with a topologically generic forcing function is dense. This gives a rigorous verification of certain numerical observations in [M. Ding, C. Grebogi and E. Ott. Evolution of attractors in quasiperiodically forced systems: from quasiperiodic to strange nonchaotic to chaotic. Phys. Rev. A 39(5) (1989), 2593–2598] for such forcing functions. More generally, under some general conditions on the base map, we show the density of the mode-locking property among dynamically forced maps (defined in [Z. Zhang. On topological genericity of the mode-locking phenomenon. Math. Ann. 376 (2020), 707–72]) equipped with a topology that is much stronger than the

$C^0$

topology, compatible with smooth fiber maps. For quasi-periodic base maps, our result generalizes the main results in [A. Avila, J. Bochi and D. Damanik. Cantor spectrum for Schrödinger operators with potentials arising from generalized skew-shifts. Duke Math. J.146 (2009), 253–280], [J. Wang, Q. Zhou and T. Jäger. Genericity of mode-locking for quasiperiodically forced circle maps. Adv. Math.348 (2019), 353–377] and Zhang (2020).

查看原文

微信好友朋友圈 QQ好友复制链接

准周期强迫阿诺德圆图的模式锁定特性密度

我们证明，具有拓扑通用强迫函数的准周期强迫阿诺德圆图族的锁模区是密集的。这就严格验证了 [M. Ding, C. Grebogi and E. Ott.Ding, C. Grebogi and E. Ott.准周期强迫系统吸引子的演化：从准周期到奇异非混沌再到混沌。Phys. Rev. A 39(5) (1989)，2593-2598] 对于这类强迫函数。更一般地说，在基图的一些一般条件下，我们展示了动态强迫图（定义见 [Z. Zhang.Zhang.On topological genericity of the mode-locking phenomenon.Math.Ann.376 (2020), 707-72]）配备的拓扑比 $C^0$ 拓扑强得多，与光滑光纤映射兼容。对于准周期基底映射，我们的结果概括了 [A. Avila, J. Bochi and J. M. Matters] 中的主要结果。Avila, J. Bochi and D. Damanik.薛定谔算子的康托谱系与广义偏移产生的势[A. Avila, J. Bochi and D. Damanik.Duke Math.J.146（2009），253-280]，[J.Wang, Q. Zhou and T. Jäger.准周期强迫圆图的锁模属性.Adv. Math.348 (2019), 353-377] 和 Zhang (2020).

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文去求助