Marat Akhmet – författare

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.Key Features

This book presents as its main subject new models in mathematical neuroscience. A wide range of neural networks models with discontinuities are discussed, including impulsive differential equations, differential equations with piecewise constant arguments, and models of mixed type. These models involve discontinuities, which are natural because huge velocities and short distances are usually observed in devices modeling the networks. A discussion of the models, appropriate for the proposed applications, is also provided.

The central subject of this book is Almost Periodic Oscillations, the most common oscillations in applications and the most intricate for mathematical analysis.  Prof. Akhmet''s lucid and rigorous examination proves these oscillations are a "regular" component of chaotic attractors. The book focuses on almost periodic functions, first of all, as Stable (asymptotically) solutions of differential equations of different types, presumably discontinuous; and, secondly, as non-isolated oscillations in chaotic sets. Finally, the author proves the existence of Almost Periodic Oscillations (asymptotic and bi-asymptotic) by asymptotic equivalence between systems. The book brings readers'' attention to contemporary methods for considering oscillations as well as to methods with strong potential for study of chaos in the future. Providing three powerful instruments for mathematical research of oscillations where dynamics are observable and applied, the book is ideal for engineers as well asspecialists in electronics, computer sciences, robotics, neural networks, artificial networks, and biology.

The book is concerned with the concepts of chaos and fractals, which are within the scopes of dynamical systems, geometry, measure theory, topology, and numerical analysis during the last several decades. It is revealed that a special kind of Poisson stable point, which we call an unpredictable point, gives rise to the existence of chaos in the quasi-minimal set. This is the first time in the literature that the description of chaos is initiated from a single motion. Chaos is now placed on the line of oscillations, and therefore, it is a subject of study in the framework of the theories of dynamical systems and differential equations, as in this book. The techniques introduced in the book make it possible to develop continuous and discrete dynamics which admit fractals as points of trajectories as well as orbits themselves. To provide strong arguments for the genericity of chaos in the real and abstract universe, the concept of abstract similarity is suggested.

Mathematical chaos in neural networks is a powerful tool that reflects the world’s complexity and has the potential to uncover the mysteries of the brain’s intellectual activity. Through this monograph, the authors aim to contribute to modern chaos research, combining it with the fundamentals of classical dynamical systems and differential equations. The readers should be reassured that an in-depth understanding of chaos theory is not a prerequisite for working in the area designed by the authors. Those interested in the discussion can have a basic understanding of ordinary differential equations and the existence of bounded solutions of quasi-linear systems on the real axis. 

Based on the novelties, this monograph aims to provide one of the most powerful approaches to studying complexities in neural networks through mathematical methods in differential equations and, consequently, to create circumstances for a deep comprehension of brain activity and artificial intelligence. A large part of the book consists of newly obtained contributions to the theory of recurrent functions, Poisson stable, and alpha unpredictable solutions and ultra Poincaré chaos of quasi-linear and strongly nonlinear neural networks such as Hopfield neural networks, shunting inhibitory cellular neural networks, inertial neural networks, and Cohen-Grossberg neural networks.

The methods and results presented in this book are meant to benefit senior researchers, engineers, and specialists working in artificial neural networks, machine and deep learning, computer science, quantum computers, and applied and pure mathematics. This broad applicability underscores the value and relevance of this research area to a large academic community and the potential impact it can have on various fields. 

This book presents detailed descriptions of chaos for continuous-time systems. It is the first-ever book to consider chaos as an input for differential and hybrid equations. Chaotic sets and chaotic functions are used as inputs for systems with attractors: equilibrium points, cycles and tori. The findings strongly suggest that chaos theory can proceed from the theory of differential equations to a higher level than previously thought. The approach selected is conducive to the in-depth analysis of different types of chaos. The appearance of deterministic chaos in neural networks, economics and mechanical systems is discussed theoretically and supported by simulations. As such, the book offers a valuable resource for mathematicians, physicists, engineers and economists studying nonlinear chaotic dynamics.