Ultra Poincar�e chaos and alpha labeling : a new approach to chaotic dynamics /
Akhmet, Marat,
Ultra Poincar�e chaos and alpha labeling : a new approach to chaotic dynamics / New approach to chaotic dynamics. Marat Akhmet. - 1 online resource (various pagings) : illustrations (some color). - [IOP release $release] IOP ebooks. [2024 collection] . - IOP (Series). Release 24. IOP ebooks. 2024 collection. .
"Version: 20241101"--Title page verso.
Includes bibliographical references.
part 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 chaos part 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. Notes 3. 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. Notes part 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. Notes part 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. Notes part 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 processes part 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. Notes part 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.
Full-text restricted to subscribers or individual document purchasers.
This 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.
Those researching or studying dynamical systems, chaos, differential equations, and stochastic differential equations.
Mode of access: World Wide Web.
System requirements: Adobe Acrobat Reader, EPUB reader, or Kindle reader.
Dr. 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.
9780750351621 9780750351614
10.1088/978-0-7503-5162-1 doi
Q172.5.C45 / A458 2024eb
Ultra Poincar�e chaos and alpha labeling : a new approach to chaotic dynamics / New approach to chaotic dynamics. Marat Akhmet. - 1 online resource (various pagings) : illustrations (some color). - [IOP release $release] IOP ebooks. [2024 collection] . - IOP (Series). Release 24. IOP ebooks. 2024 collection. .
"Version: 20241101"--Title page verso.
Includes bibliographical references.
part 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 chaos part 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. Notes 3. 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. Notes part 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. Notes part 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. Notes part 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 processes part 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. Notes part 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.
Full-text restricted to subscribers or individual document purchasers.
This 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.
Those researching or studying dynamical systems, chaos, differential equations, and stochastic differential equations.
Mode of access: World Wide Web.
System requirements: Adobe Acrobat Reader, EPUB reader, or Kindle reader.
Dr. 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.
9780750351621 9780750351614
10.1088/978-0-7503-5162-1 doi
Q172.5.C45 / A458 2024eb