Domain Structured Dynamics : Unpredictability, Chaos, Randomness, Fractals, Differential Equations and Neural Networks / Marat Akhmet.

Format:

Book

Author/Creator:

Akhmet, Marat, author.

Contributor:

Institute of Physics (Great Britain), publisher.

Series:

IOP ebooks. 2021 collection.

IOP Ebooks Series

Language:

English

Subjects (All):

Chaotic behavior in systems.

Mathematical physics.

Difference equations.

Physical Description:

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

Edition:

First edition.

Place of Publication:

Bristol [England] (Temple Circus, Temple Way, Bristol BS1 6HG, UK) : IOP Publishing, [2021]

System Details:

Mode of access: World Wide Web.

System requirements: Adobe Acrobat Reader, EPUB reader, or Kindle reader.

Biography/History:

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.

Summary:

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.

Contents:

3. Domain structured dynamics : theory

3.1. Basic definitions and theorems

3.2. Hyperbolic chaos

3.3. Modular chaos

3.4. Abstract fractals

3.5. Notes

4. 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. Notes

part 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 processes

part 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. Notes

part 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.

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 networks

part II. Chaos and fractals. 2. Unpredictability implies chaos

2.1. Unpredictability in dynamics

2.2. Unpredictable functions

2.3. Notes

Notes:

"Version: 202108"--Title page verso.

Includes bibliographical references.

Title from PDF title page (viewed on September 1, 2021).

Description based on print version record.

ISBN:

9780750335065

0750335068

9780750335072

0750335076

OCLC:

1266548340