{"title":"混沌一维映射精确轨迹解的施罗德形式表示","authors":"Valerij Anikin","doi":"10.18500/0869-6632-003034","DOIUrl":null,"url":null,"abstract":"Purpose of the article is to illustrate the genesis, meaning and significance of the functional Schroder equation, introduced in the theory of iterations of rational functions, for the theory of deterministic chaos by analytical calculations of exact trajectory solutions, invariant densities and Lyapunov exponents of one-dimensional chaotic maps. We demonstrate the method for solving the functional Schroder equation for various chaotic maps by passing to a topologically conjugate mappings, for which finding the exact trajectory solution is a simpler mathematical procedure. Results of the analytical solution of the Schroder equation for 12 chaotic mappings of various types and the calculation of the corresponding expressions for exact trajectory solutions, invariant densities and Lyapunov exponents are presented. Conclusion is made about the methodological expediency of formulating and solving the Schroder equations by the study of the dynamics of one-dimensional chaotic mappings.","PeriodicalId":41611,"journal":{"name":"Izvestiya Vysshikh Uchebnykh Zavedeniy-Prikladnaya Nelineynaya Dinamika","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Representation of exact trajectory solutions for chaotic one-dimensional maps in Schroder form\",\"authors\":\"Valerij Anikin\",\"doi\":\"10.18500/0869-6632-003034\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Purpose of the article is to illustrate the genesis, meaning and significance of the functional Schroder equation, introduced in the theory of iterations of rational functions, for the theory of deterministic chaos by analytical calculations of exact trajectory solutions, invariant densities and Lyapunov exponents of one-dimensional chaotic maps. We demonstrate the method for solving the functional Schroder equation for various chaotic maps by passing to a topologically conjugate mappings, for which finding the exact trajectory solution is a simpler mathematical procedure. Results of the analytical solution of the Schroder equation for 12 chaotic mappings of various types and the calculation of the corresponding expressions for exact trajectory solutions, invariant densities and Lyapunov exponents are presented. Conclusion is made about the methodological expediency of formulating and solving the Schroder equations by the study of the dynamics of one-dimensional chaotic mappings.\",\"PeriodicalId\":41611,\"journal\":{\"name\":\"Izvestiya Vysshikh Uchebnykh Zavedeniy-Prikladnaya Nelineynaya Dinamika\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2023-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Izvestiya Vysshikh Uchebnykh Zavedeniy-Prikladnaya Nelineynaya Dinamika\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.18500/0869-6632-003034\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"PHYSICS, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Izvestiya Vysshikh Uchebnykh Zavedeniy-Prikladnaya Nelineynaya Dinamika","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.18500/0869-6632-003034","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"PHYSICS, MULTIDISCIPLINARY","Score":null,"Total":0}
Representation of exact trajectory solutions for chaotic one-dimensional maps in Schroder form
Purpose of the article is to illustrate the genesis, meaning and significance of the functional Schroder equation, introduced in the theory of iterations of rational functions, for the theory of deterministic chaos by analytical calculations of exact trajectory solutions, invariant densities and Lyapunov exponents of one-dimensional chaotic maps. We demonstrate the method for solving the functional Schroder equation for various chaotic maps by passing to a topologically conjugate mappings, for which finding the exact trajectory solution is a simpler mathematical procedure. Results of the analytical solution of the Schroder equation for 12 chaotic mappings of various types and the calculation of the corresponding expressions for exact trajectory solutions, invariant densities and Lyapunov exponents are presented. Conclusion is made about the methodological expediency of formulating and solving the Schroder equations by the study of the dynamics of one-dimensional chaotic mappings.
期刊介绍：
Scientific and technical journal Izvestiya VUZ. Applied Nonlinear Dynamics is an original interdisciplinary publication of wide focus. The journal is included in the List of periodic scientific and technical publications of the Russian Federation, recommended for doctoral thesis publications of State Commission for Academic Degrees and Titles at the Ministry of Education and Science of the Russian Federation, indexed by Scopus, RSCI. The journal is published in Russian (English articles are also acceptable, with the possibility of publishing selected articles in other languages by agreement with the editors), the articles data as well as abstracts, keywords and references are consistently translated into English. First and foremost the journal publishes original research in the following areas: -Nonlinear Waves. Solitons. Autowaves. Self-Organization. -Bifurcation in Dynamical Systems. Deterministic Chaos. Quantum Chaos. -Applied Problems of Nonlinear Oscillation and Wave Theory. -Modeling of Global Processes. Nonlinear Dynamics and Humanities. -Innovations in Applied Physics. -Nonlinear Dynamics and Neuroscience. All articles are consistently sent for independent, anonymous peer review by leading experts in the relevant fields, the decision to publish is made by the Editorial Board and is based on the review. In complicated and disputable cases it is possible to review the manuscript twice or three times. The journal publishes review papers, educational papers, related to the history of science and technology articles in the following sections: -Reviews of Actual Problems of Nonlinear Dynamics. -Science for Education. Methodical Papers. -History of Nonlinear Dynamics. Personalia.
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