06928nam a2200697 i 45000010014000000030004000140050017000180060019000350070015000540080041000690200025001100200024001350200025001590200027001840240035002110350025002460350022002710400040002930500028003330720017003610720023003781000035004012450102004362460038005382640098005763000071006743360021007453370026007663380032007924900027008244900034008515000043008855040041009285050918009695050351018875050247022385050277024855050673027625050270034355050218037055050254039235060075041775200881042525210123051335300029052565380036052855380078053215450331053995880061057306500040057916500021058316500033058526500068058857100061059537760049060148300038060638300041061018560059061429420012062019990017062139780750351621IOP20251030124828.0m    eo  d        cr cn |||m|||a241213s2024    enka   fob    000 0 eng d  a9780750351621qebook  a9780750351614qmobi  z9780750351607qprint  z9780750351638qmyPrint7 a10.1088/978-0-7503-5162-12doi  a(CaBNVSL)thg00084069  a(OCoLC)1475037933  aCaBNVSLbengerdacCaBNVSLdCaBNVSL 4aQ172.5.C45bA458 2024eb 7aPBWS2bicssc 7aMAT0070002bisacsh1 aAkhmet, Marat,eauthor.9969371 aUltra Poincar�e chaos and alpha labeling :ba new approach to chaotic dynamics /cMarat Akhmet.3 aNew approach to chaotic dynamics. 1aBristol [England] (Temple Circus, Temple Way, Bristol BS1 6HG, UK) :bIOP Publishing,c[2024]  a1 online resource (various pagings) :billustrations (some color).  atext2rdacontent  aelectronic2isbdmedia  aonline resource2rdacarrier1 a[IOP release $release]1 aIOP ebooks. [2024 collection]  a"Version: 20241101"--Title page verso.  aIncludes bibliographical references.  apart I. Introduction. 1. Historical and philosophical overview -- 1.1. Outline -- 1.2. A new mathematical structure -- 1.3. Alpha labels in spiraled hierarchy of mathematical structures -- 1.4. Universality and minimalism of alpha labeling and unpredictability -- 1.5. Principle of excessiveness : the Universe approximates science -- 1.6. Potential and actual uncertainty in dynamics -- 1.7. Actual uncertainty of chaos -- 1.8. Monads spiral up to abstract similarity -- 1.9. Alpha labeling for more chaos and fractals -- 1.10. Ancient Greek chaos -- 1.11. Alpha unpredictability and alpha labeling decrease formalism of chaos -- 1.12. Mathematical chaos and H. Poincar�e -- 1.13. Alpha unpredictability versus sensitivity -- 1.14. Ultra Poincar�e chaos versus Li-Yorke chaos -- 1.15. Chaos as a metaphor -- 1.16. Does the theory of chaos exist? -- 1.17. Ubiquitous chaos and dualism -- 1.18. Domain of chaos8 apart II. Alpha labeling and unpredictability. 2. Alpha labels are a new mathematical structure -- 2.1. Sets of alpha labels and alpha map -- 2.2. Alpha dynamics -- 2.3. Alpha chaos -- 2.4. Examples of alpha chaotic models -- 2.5. Hyperbolic alpha labeling -- 2.6. Modular chaos -- 2.7. Abstract fractals : alpha spaces with measures -- 2.8. Notes8 a3. Alpha unpredictability implies a universal mathematical chaos -- 3.1. Poincar�e and Lorenz lines in a nutshell -- 3.2. An individual chaos : alpha unpredictability -- 3.3. Alpha unpredictable functions : appearance of chaos -- 3.4. Notes8 apart III. Compartmental functions. 4. Compartmental alpha unpredictable functions -- 4.1. Continuous compartmental unpredictable functions -- 4.2. Discontinuous compartmental periodic Poisson stable functions -- 4.3. Algebra and significance for applications -- 4.4. Notes8 apart IV. Differential equations with ultra Poincar�e chaos. 5. Alpha unpredictable differential equations -- 5.1. Strongly alpha unpredictable solutions -- 5.2. Compartmental quasi-linear equations -- 5.3. Modulo periodic Poisson stable solutions -- part V. Numerical alpha unpredictability. 6. Ultra Poincar�e chaos numerically -- 6.1. Introduction and preliminaries -- 6.2. The algorithm of the test and the application procedure -- 6.3. Ultra Poincar�e chaos for revisited models -- 6.4. Alpha unpredictability test is positive for known strange non-chaotic attractors -- 6.5. The generalized synchronization as a proof of alpha unpredictability -- 6.6. Notes8 apart VI. Randomness and alpha labeling. 7. Alpha labeled randomness -- 7.1. Alpha unpredictability in Bernoulli schemes -- 7.2. Alpha unpredictable functions randomly -- 7.3. Alpha unpredictable strings and statistical laws -- 7.4. Modular chaos in random processes8 apart VII. Markov chains and differential equations. 8. Markov chains and stochastic differential equations -- 8.1. Alpha chaos in Markov chains -- 8.2. Duffing type equations with Markov coefficients -- 8.3. Notes8 apart VIII. Fractals with alpha spaces. 9. Alpha induced dynamics in fractals and cubes -- 9.1. Historical observations -- 9.2. Alpha induced dynamics in metric spaces -- 9.3. Chaotic cubes -- 9.4. Alpha chaos in Fatou-Julia iterations -- 9.5. Notes.  aFull-text restricted to subscribers or individual document purchasers.3 aThis book serves as a comprehensive and detailed collection of knowledge on two innovative aspects of our research: ultra Poincar�e chaos and alpha labeling. The first concept represents a fundamental trait of dynamical complexity, while the second acts as an algorithmic tool to analyse and construct complexity within the dynamics, probabilities, and geometries of science and industry. The manuscript aims to provide solid guidance for studying complexities through rigorous mathematical methods. The chaos is derived from the dynamical characteristic of alpha unpredictability, representing a modernized version of Poisson stability motion. It builds upon the foundational work of Poincar�e and Birkhoff, incorporating key new insights that expand upon the French genius's contributions to the recurrence theorem, applicable in contexts such as the three-body problem.  aThose researching or studying dynamical systems, chaos, differential equations, and stochastic differential equations.  aAlso available in print.  aMode of access: World Wide Web.  aSystem requirements: Adobe Acrobat Reader, EPUB reader, or Kindle reader.  aDr. Marat Akhmet is presently a Professor in the Department of Mathematics at METU in Ankara, Turkey. In recent years, he has been exploring continuous and discontinuous dynamics, artificial neural networks alongside stability, chaos, and fractals. He has authored eight books and published over two hundred scientific papers.  aTitle from PDF title page (viewed on December 13, 2024).  aChaotic behavior in systems.996938  aDynamics.965524 7aChaos theory.2bicssc996939 7aMATHEMATICS / Differential Equations / General.2bisacsh9969402 aInstitute of Physics (Great Britain),epublisher.917702 8iPrint version:z9780750351607z9780750351638 0aIOP (Series).pRelease 24.996941  aIOP ebooks.p2024 collection.9969424 uhttps://iopscience.iop.org/book/mono/978-0-7503-5162-1  cE-Books  c68468d68468