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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.
Marat Akhmet is currently a Professor of Mathematics at Middle East Technical University in Ankara, Türkiye. He received his bachelor's degree in Mathematics from Aktobe State University in Kazakhstan and his Doctorate in Differential Equations and Mathematical Physics from Kyiv State University in Ukraine. M. Akhmet has been awarded the Science Prize of TUBITAK (Türkiye, 2015) for his achievements in scientific research and the Prof C.S. Hsu Award (UC Berkeley, 2021) for outstanding results in non-linear analysis investigations. Has been invited as a Plenary speaker for 17 international conferences in the UK, Kazakhstan, Türkiye, the USA, France, and Italy, and organized more than 20 international and national scientific meetings. Dr. Akhmet is passionate about scientific research and his role as a teacher, and he has continuously been one of the recipients of the "Best Performance Award" at Middle East Technical University over the 20 years of his work there. Dr. Akhmet has mentored 20 Ph.D. students over his career, who now work in research centers and universities in Türkiye, Kazakhstan, and Libya. Dr. Marat Akhmet's research focuses on dynamical models and differential equations. He has published seven Springer books and more than 150 scientific papers. In the last several years, he has been investigating the dynamics of neural networks, almost periodic motions, stability, synchronization, chaos, and fractals.
Madina Tleubergenova is an Associate professor of Mathematics, working as the head of the Mathematics department at K. Zhubanov Aktobe Regional University. She received her Bachelor's degree in Mathematics at K. Zhubanov Aktobe State University and her Doctoral degree in Physical and Mathematical Sciences at Al-Farabi Kazakh National University, Kazakhstan. She is the head of research on fundamental research projects of the Ministry of Higher Education and Science of the Republic of Kazakhstan, and the winner of the "Best University Teacher 2020" prize awarded by the Republic of Kazakhstan.
Akylbek Zhamanshin is an Associate professor of Mathematics at the Herriot-Watt International Faculty of Aktobe Regional University, Aktobe, Kazakhstan. He received his PhD degree in Mathematics at Aktobe Regional University and was a postdoctoral researcher at Middle East Technical University, Ankara, Türkiye. His research interests include differential equations, dynamical systems, complex dynamics, neural networks, and chaos.
Zakhira Nugayeva is a Mathematics teacher at the Department of Mathematics of Aktobe Regional University, Aktobe, Kazakhstan. She received her PhD degree in Mathematics at Aktobe Regional University. Her research interests include differential equations, impulsive systems, complex dynamics, neural networks, and chaos. She won the title "Best University Teacher 2023" in Kazakhstan.
Preface.- 1. Introduction.- 2. Preliminaries.- 3. Hopfield-type neural networks.- 4. Shunting inhibitory cellular neural networks.- 5. Inertial neural networks with discontinuities.- 6. Cohen-Grossberg neural networks.
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