{"title":"分数达芬方程中丰富的解现象学","authors":"Sara Hamaizia, Salvador Jiménez, M. Pilar Velasco","doi":"10.1007/s13540-024-00269-1","DOIUrl":null,"url":null,"abstract":"<p>In this paper, we characterize the chaos in the Duffing equation with negative linear stiffness and a fractional damping term given by a Caputo fractional derivative of order <span>\\(\\alpha \\)</span> ranging from 0 to 2. We use two different numerical methods to compute the solutions, one of them new. We discriminate between regular and chaotic solutions by means of the attractor in the phase space and the values of the Lyapunov Characteristic Exponents. For this, we have extended a linear approximation method to this equation. The system is very rich with distinct behaviours. In the limits <span>\\(\\alpha \\)</span> to 0 or <span>\\(\\alpha \\)</span> to 2, the system tends to basically the same undamped system with a behaviour clearly different from the classical Duffing equation.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"24 1","pages":""},"PeriodicalIF":2.5000,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Rich phenomenology of the solutions in a fractional Duffing equation\",\"authors\":\"Sara Hamaizia, Salvador Jiménez, M. Pilar Velasco\",\"doi\":\"10.1007/s13540-024-00269-1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>In this paper, we characterize the chaos in the Duffing equation with negative linear stiffness and a fractional damping term given by a Caputo fractional derivative of order <span>\\\\(\\\\alpha \\\\)</span> ranging from 0 to 2. We use two different numerical methods to compute the solutions, one of them new. We discriminate between regular and chaotic solutions by means of the attractor in the phase space and the values of the Lyapunov Characteristic Exponents. For this, we have extended a linear approximation method to this equation. The system is very rich with distinct behaviours. In the limits <span>\\\\(\\\\alpha \\\\)</span> to 0 or <span>\\\\(\\\\alpha \\\\)</span> to 2, the system tends to basically the same undamped system with a behaviour clearly different from the classical Duffing equation.</p>\",\"PeriodicalId\":48928,\"journal\":{\"name\":\"Fractional Calculus and Applied Analysis\",\"volume\":\"24 1\",\"pages\":\"\"},\"PeriodicalIF\":2.5000,\"publicationDate\":\"2024-03-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Fractional Calculus and Applied Analysis\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s13540-024-00269-1\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Fractional Calculus and Applied Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s13540-024-00269-1","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Rich phenomenology of the solutions in a fractional Duffing equation
In this paper, we characterize the chaos in the Duffing equation with negative linear stiffness and a fractional damping term given by a Caputo fractional derivative of order \(\alpha \) ranging from 0 to 2. We use two different numerical methods to compute the solutions, one of them new. We discriminate between regular and chaotic solutions by means of the attractor in the phase space and the values of the Lyapunov Characteristic Exponents. For this, we have extended a linear approximation method to this equation. The system is very rich with distinct behaviours. In the limits \(\alpha \) to 0 or \(\alpha \) to 2, the system tends to basically the same undamped system with a behaviour clearly different from the classical Duffing equation.
期刊介绍:
Fractional Calculus and Applied Analysis (FCAA, abbreviated in the World databases as Fract. Calc. Appl. Anal. or FRACT CALC APPL ANAL) is a specialized international journal for theory and applications of an important branch of Mathematical Analysis (Calculus) where differentiations and integrations can be of arbitrary non-integer order. The high standards of its contents are guaranteed by the prominent members of Editorial Board and the expertise of invited external reviewers, and proven by the recently achieved high values of impact factor (JIF) and impact rang (SJR), launching the journal to top places of the ranking lists of Thomson Reuters and Scopus.