Domain structured dynamics :unpredictability, chaos, randomness, fractals, differential equations and neural networks /

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"Version: 202108"--Title page verso.Includes bibliographical references.part I. Introduction. 1. What this book is about -- 1.1. Unpredictability -- 1.2. Domain structured dynamics -- 1.3. Random processes -- 1.4. Abstract fractals -- 1.5. Differential equations -- 1.6. Neural networkspart II. Chaos and fractals. 2. Unpredictability implies chaos -- 2.1. Unpredictability in dynamics -- 2.2. Unpredictable functions -- 2.3. Notes3. Domain structured dynamics : theory -- 3.1. Basic definitions and theorems -- 3.2. Hyperbolic chaos -- 3.3. Modular chaos -- 3.4. Abstract fractals -- 3.5. Notes4. Domain structured dynamics : applications -- 4.1. Theoretical preliminaries -- 4.2. Chaotic finite-dimensional cubes -- 4.3. Similarity through Fatou-Julia iterations -- 4.4. Conservative-progressive growth -- 4.5. Domain structured dynamics in fractals -- 4.6. Notespart III. Random processes. 5. Randomness with similarity dynamics -- 5.1. The basics of abstract similarity -- 5.2. Unpredictability in Bernoulli schemes -- 5.3. Random unpredictable functions -- 5.4. Unpredictable strings -- 5.5. Abstract similarity chaos in Markov chains -- 5.6. Modular chaos in random processespart IV. Non-autonomous differential equations. 6. Unpredictability in differential equations -- 6.1. Preliminaries -- 6.2. Strongly unpredictable solutions -- 6.3. Unpredictable solutions -- 6.4. Examples -- 6.5. Notespart V. Neural networks. 7. Unpredictable oscillations of neural networks -- 7.1. SICNNs with strongly unpredictable oscillations -- 7.2. Inertial neural networks with unpredictable oscillations -- 7.3. Notes.Domain structured dynamics introduces a way for analysis of chaos in fractals, neural networks and random processes. It starts with newly invented abstract similarity sets and maps, which are in the basis of the abstract similarity dynamics. Then a labeling procedure is designed to determine the domain structured dynamics. The results follow the Pythagorean doctrine, considering finite number of indices for the labeling, with potential to become universal in future. The immediate power of the approach for fractals as domains of chaos, revisited famous deterministic and stochastic models, new types of differential equations and neural networks is seen in the book. This is not considered through widening areas, where the notions can be seen and recognized, but by deepening abstraction.Specialists in nonlinear dynamics, differential and discrete equations, chaos theory, fractal geometry and applications. Engineers, experts in industrial development of all kinds: electronics, chemistry, mechanics, control and optimization.Also available in print.Mode of access: World Wide Web.System requirements: Adobe Acrobat Reader, EPUB reader, or Kindle reader.Dr. Marat Akhmet is currently a Professor at Department of Mathematics, METU, Ankara, Turkey. He received his PhD in differential equations and mathematical physics at Kiev State University, Ukraine. Marat Akhmet's research focuses on the dynamical models and differential equations. He has published seven books and more than a hundred and fifty scientific papers. In the last several years, he has been investigating dynamics of neural networks, periodic, almost periodic and unpredictable motions, stability, chaos, and fractals.Title from PDF title page (viewed on September 1, 2021).

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: .,

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1 online resource (various pagings) :illustrations (some color).

Language

English

ISBN/ISSN

9780750335072

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003/.857

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Subject(s)

Mathematics and computation.
Mathematical physics.
Dynamics.
Chaotic behavior in systems.
Nonlinear theories.
Difference equations.
Fractals.

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Marat Akhmet.

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