Marat Akhmet's Books

Dynamics with Chaos and Fractals

By: Marat Akhmet,Mehmet Onur Fen,Ejaily Milad Alejaily

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Dynamics with Chaos and Fractals

By: Marat Akhmet,Mehmet Onur Fen,Ejaily Milad Alejaily

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.

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Page Count

233

Publisher

Springer Nature

Published Date

ISBN 10

3030358542

ISBN 13

9783030358549

Replication of Chaos in Neural Networks, Economics and Physics

By: Marat Akhmet,Mehmet Onur Fen

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Replication of Chaos in Neural Networks, Economics and Physics

By: Marat Akhmet,Mehmet Onur Fen

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.

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Page Count

468

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Springer

Published Date

ISBN 10

3662475006

ISBN 13

9783662475003

Almost Periodicity, Chaos, and Asymptotic Equivalence

By: Marat Akhmet

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Almost Periodicity, Chaos, and Asymptotic Equivalence

By: Marat Akhmet

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 as specialists in electronics, computer sciences, robotics, neural networks, artificial networks, and biology. Distinctively combines results and methods of the theory of differential equations with thorough investigation of chaotic dynamics with almost periodic ingredients; Provides all necessary mathematical basics in their most developed form, negating the need for any additional sources for readers to start work in the area; Presents a unique method of investigation of discontinuous almost periodic solutions in its unified form, employed to differential equations with different types of discontinuity; Develops the equivalence method to its ultimate effective state such that most important theoretical problems and practical applications can be analyzed by the method.

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Page Count

368

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Springer

Published Date

ISBN 10

303020572X

ISBN 13

9783030205720

Nonlinear Hybrid Continuous/Discrete-Time Models

By: Marat Akhmet

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Nonlinear Hybrid Continuous/Discrete-Time Models

By: Marat Akhmet

The book is mainly about hybrid systems with continuous/discrete-time dynamics. The major part of the book consists of the theory of equations with piece-wise constant argument of generalized type. The systems as well as technique of investigation were introduced by the author very recently. They both generalized known theory about differential equations with piece-wise constant argument, introduced by K. Cook and J. Wiener in the 1980s. Moreover, differential equations with fixed and variable moments of impulses are used to model real world problems. We consider models of neural networks, blood pressure distribution and a generalized model of the cardiac pacemaker. All the results of the manuscript have not been published in any book, yet. They are very recent and united with the presence of the continuous/discrete dynamics of time. It is of big interest for specialists in biology, medicine, engineering sciences, electronics. Theoretical aspects of the book meet very strong expectations of mathematicians who investigate differential equations with discontinuities of any type.

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Page Count

225

Publisher

Springer Science & Business Media

Published Date

ISBN 10

9491216031

ISBN 13

9789491216039

Neural Networks with Discontinuous/Impact Activations

By: Marat Akhmet,Enes Yılmaz

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Neural Networks with Discontinuous/Impact Activations

By: Marat Akhmet,Enes Yılmaz

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.

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176

Publisher

Springer Science & Business Media

Published Date

ISBN 10

1461485665

ISBN 13

9781461485667

Principles of Discontinuous Dynamical Systems

By: Marat Akhmet

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Principles of Discontinuous Dynamical Systems

By: Marat Akhmet

Discontinuous dynamical systems have played an important role in both theory and applications during the last several decades. This is still an area of active research and techniques to make the applications more effective are an ongoing topic of interest. Principles of Discontinuous Dynamical Systems is devoted to the theory of differential equations with variable moments of impulses. It introduces a new strategy of implementing an equivalence to systems whose solutions have prescribed moments of impulses and utilizing special topologies in spaces of piecewise continuous functions. The achievements obtained on the basis of this approach are described in this book. The text progresses systematically, by covering preliminaries in the first four chapters. This is followed by more complex material and special topics such as Hopf bifurcation, Devaney's chaos, and the shadowing property are discussed in the last two chapters. This book is suitable for researchers and graduate students in mathematics and also in diverse areas such as biology, computer science, and engineering who deal with real world problems.

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Page Count

185

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Springer Science & Business Media

Published Date

ISBN 10

1441965815

ISBN 13

9781441965813

Almost Periodicity, Chaos, and Asymptotic Equivalence

By: Marat Akhmet

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Almost Periodicity, Chaos, and Asymptotic Equivalence

By: Marat Akhmet

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Page Count

368

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Springer

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ISBN 10

303020572X

ISBN 13

9783030205720

Replication of Chaos in Neural Networks, Economics and Physics

By: Marat Akhmet,Mehmet Onur Fen

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Replication of Chaos in Neural Networks, Economics and Physics

By: Marat Akhmet,Mehmet Onur Fen

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Page Count

468

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Springer

Published Date

ISBN 10

3662475006

ISBN 13

9783662475003

Principles of Discontinuous Dynamical Systems

By: Marat Akhmet

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Principles of Discontinuous Dynamical Systems

By: Marat Akhmet

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185

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Springer Science & Business Media

Published Date

ISBN 10

1441965815

ISBN 13

9781441965813

Dynamics with Chaos and Fractals

By: Marat Akhmet,Mehmet Onur Fen,Ejaily Milad Alejaily

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Dynamics with Chaos and Fractals

By: Marat Akhmet,Mehmet Onur Fen,Ejaily Milad Alejaily

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Page Count

233

Publisher

Springer Nature

Published Date

ISBN 10

3030358542

ISBN 13

9783030358549

Nonlinear Hybrid Continuous/Discrete-Time Models

By: Marat Akhmet

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Nonlinear Hybrid Continuous/Discrete-Time Models

By: Marat Akhmet

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Page Count

225

Publisher

Springer Science & Business Media

Published Date

ISBN 10

9491216031

ISBN 13

9789491216039

Neural Networks with Discontinuous/Impact Activations

By: Marat Akhmet,Enes Yılmaz

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Neural Networks with Discontinuous/Impact Activations

By: Marat Akhmet,Enes Yılmaz

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Page Count

176

Publisher

Springer Science & Business Media

Published Date

ISBN 10

1461485665

ISBN 13

9781461485667

Bifurcation in Autonomous and Nonautonomous Differential Equations with Discontinuities

By: Marat Akhmet,Ardak Kashkynbayev

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Bifurcation in Autonomous and Nonautonomous Differential Equations with Discontinuities

By: Marat Akhmet,Ardak Kashkynbayev

This book focuses on bifurcation theory for autonomous and nonautonomous differential equations with discontinuities of different types – those with jumps present either in the right-hand side, or in trajectories or in the arguments of solutions of equations. The results obtained can be applied to various fields, such as neural networks, brain dynamics, mechanical systems, weather phenomena and population dynamics. Developing bifurcation theory for various types of differential equations, the book is pioneering in the field. It presents the latest results and provides a practical guide to applying the theory to differential equations with various types of discontinuity. Moreover, it offers new ways to analyze nonautonomous bifurcation scenarios in these equations. As such, it shows undergraduate and graduate students how bifurcation theory can be developed not only for discrete and continuous systems, but also for those that combine these systems in very different ways. At the same time, it offers specialists several powerful instruments developed for the theory of discontinuous dynamical systems with variable moments of impact, differential equations with piecewise constant arguments of generalized type and Filippov systems.

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Page Count

175

Publisher

Springer

Published Date

ISBN 10

9811031800

ISBN 13

9789811031809

Book's by Marat Akhmet

Dynamics with Chaos and Fractals

Dynamics with Chaos and Fractals

Replication of Chaos in Neural Networks, Economics and Physics

Replication of Chaos in Neural Networks, Economics and Physics

Almost Periodicity, Chaos, and Asymptotic Equivalence

Almost Periodicity, Chaos, and Asymptotic Equivalence

Nonlinear Hybrid Continuous/Discrete-Time Models

Nonlinear Hybrid Continuous/Discrete-Time Models

Neural Networks with Discontinuous/Impact Activations

Neural Networks with Discontinuous/Impact Activations

Principles of Discontinuous Dynamical Systems

Principles of Discontinuous Dynamical Systems

Almost Periodicity, Chaos, and Asymptotic Equivalence

Almost Periodicity, Chaos, and Asymptotic Equivalence

Replication of Chaos in Neural Networks, Economics and Physics

Replication of Chaos in Neural Networks, Economics and Physics

Principles of Discontinuous Dynamical Systems

Principles of Discontinuous Dynamical Systems

Dynamics with Chaos and Fractals

Dynamics with Chaos and Fractals

Nonlinear Hybrid Continuous/Discrete-Time Models

Nonlinear Hybrid Continuous/Discrete-Time Models

Neural Networks with Discontinuous/Impact Activations

Neural Networks with Discontinuous/Impact Activations

Bifurcation in Autonomous and Nonautonomous Differential Equations with Discontinuities

Bifurcation in Autonomous and Nonautonomous Differential Equations with Discontinuities

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