Strange attractor of the Lozi mappings for the parameter region [0 IF 1.2 3区 数学 Q1 MATHEMATICS Journal of Mathematical Analysis and Applications Pub Date : 2024-10-31 DOI:10.1016/j.jmaa.2024.129018 Khadija Ben Rejeb 求助PDF {"title":"Strange attractor of the Lozi mappings for the parameter region [0<b<1,b+1<a<2−b2]","authors":"Khadija Ben Rejeb","doi":"10.1016/j.jmaa.2024.129018","DOIUrl":null,"url":null,"abstract":"<div><div>In this paper, we give a mathematical proof to the existence of a strange attractor for the Lozi mapping <em>L</em>. More precisely, we prove that <em>L</em> has a unique strange attractor for the parameter region [<span><math><mn>0</mn><mo>&lt;</mo><mi>b</mi><mo>&lt;</mo><mn>1</mn><mo>,</mo><mspace></mspace><mi>b</mi><mo>+</mo><mn>1</mn><mo>&lt;</mo><mi>a</mi><mo>&lt;</mo><mn>2</mn><mo>−</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>] which coincides with the closure of the unstable manifold at the fixed point <span><math><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mo>+</mo><mi>a</mi><mo>−</mo><mi>b</mi></mrow></mfrac><mo>,</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mn>1</mn><mo>+</mo><mi>a</mi><mo>−</mo><mi>b</mi></mrow></mfrac><mo>)</mo></math></span>. This extends a result obtained by (M. Misiurewicz, Strange attractor for the Lozi mapping, Ann.N.Y. Acad. Sci. 357, (1980), pp. 348-358). On the other hand, we study the dynamical behavior of the map <em>L</em> on its strange attractor and we prove that it is Li-Yorke chaotic. MSC 2010 Primary: 37D45, 37E30.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"543 2","pages":"Article 129018"},"PeriodicalIF":1.2000,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Analysis and Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022247X24009405","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0} 引用次数: 0 引用 批量引用 Abstract In this paper, we give a mathematical proof to the existence of a strange attractor for the Lozi mapping L. More precisely, we prove that L has a unique strange attractor for the parameter region [0<b<1,b+1<a<2−b2] which coincides with the closure of the unstable manifold at the fixed point (11+a−b,b1+a−b). This extends a result obtained by (M. Misiurewicz, Strange attractor for the Lozi mapping, Ann.N.Y. Acad. Sci. 357, (1980), pp. 348-358). On the other hand, we study the dynamical behavior of the map L on its strange attractor and we prove that it is Li-Yorke chaotic. MSC 2010 Primary: 37D45, 37E30. 分享 查看原文 本刊更多论文 参数区域 [0 更确切地说，我们证明了洛齐映射 L 在参数区域 [0<b<1,b+1<a<2-b2] 有一个唯一的奇异吸引子，它与不稳定流形在定点 (11+a-b,b1+a-b) 的闭合重合。这扩展了 (M. Misiurewicz, Strange attractor for the Lozi mapping, Ann.N.Y. Acad.357, (1980), pp.）另一方面，我们研究了映射 L 在其奇异吸引子上的动力学行为，并证明它是李-约克混沌的。MSC 2010 Primary: 37D45, 37E30. 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 求助全文 来源期刊 Journal of Mathematical Analysis and Applications 数学-数学 CiteScore 2.50 自引率 7.70% 发文量 790 审稿时长 6 months 期刊介绍： The Journal of Mathematical Analysis and Applications presents papers that treat mathematical analysis and its numerous applications. The journal emphasizes articles devoted to the mathematical treatment of questions arising in physics, chemistry, biology, and engineering, particularly those that stress analytical aspects and novel problems and their solutions. 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Abstract

In this paper, we give a mathematical proof to the existence of a strange attractor for the Lozi mapping L. More precisely, we prove that L has a unique strange attractor for the parameter region [$0<b<1,b+1<a<2-\frac{b}{2}$] which coincides with the closure of the unstable manifold at the fixed point $(\frac{1}{1+a-b},\frac{b}{1+a-b})$. This extends a result obtained by (M. Misiurewicz, Strange attractor for the Lozi mapping, Ann.N.Y. Acad. Sci. 357, (1980), pp. 348-358). On the other hand, we study the dynamical behavior of the map L on its strange attractor and we prove that it is Li-Yorke chaotic. MSC 2010 Primary: 37D45, 37E30.