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Stability Theory for Dynamic Equations on Time Scales

Anatoly A. Martynyuk

Author : Anatoly A. Martynyuk
Publisher : Birkhäuser
Page : 223 pages
File Size : 44,8 Mb
Release : 2016-09-22
Category : Mathematics
ISBN : 9783319422138

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Stability Theory for Dynamic Equations on Time Scales by Anatoly A. Martynyuk Pdf

This monograph is a first in the world to present three approaches for stability analysis of solutions of dynamic equations. The first approach is based on the application of dynamic integral inequalities and the fundamental matrix of solutions of linear approximation of dynamic equations. The second is based on the generalization of the direct Lyapunovs method for equations on time scales, using scalar, vector and matrix-valued auxiliary functions. The third approach is the application of auxiliary functions (scalar, vector, or matrix-valued ones) in combination with differential dynamic inequalities. This is an alternative comparison method, developed for time continuous and time discrete systems.In recent decades, automatic control theory in the study of air- and spacecraft dynamics and in other areas of modern applied mathematics has encountered problems in the analysis of the behavior of solutions of time continuous-discrete linear and/or nonlinear equations of perturbed motion. In the book “Men of Mathematics,” 1937, E.T.Bell wrote: “A major task of mathematics today is to harmonize the continuous and the discrete, to include them in one comprehensive mathematics, and to eliminate obscurity from both.”Mathematical analysis on time scales accomplishes exactly this. This research has potential applications in such areas as theoretical and applied mechanics, neurodynamics, mathematical biology and finance among others.

Dynamic Equations on Time Scales

Martin Bohner,Allan Peterson

Author : Martin Bohner,Allan Peterson
Publisher : Springer Science & Business Media
Page : 365 pages
File Size : 46,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461202011

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Dynamic Equations on Time Scales by Martin Bohner,Allan Peterson Pdf

On becoming familiar with difference equations and their close re lation to differential equations, I was in hopes that the theory of difference equations could be brought completely abreast with that for ordinary differential equations. [HUGH L. TURRITTIN, My Mathematical Expectations, Springer Lecture Notes 312 (page 10), 1973] A major task of mathematics today is to harmonize the continuous and the discrete, to include them in one comprehensive mathematics, and to eliminate obscurity from both. [E. T. BELL, Men of Mathematics, Simon and Schuster, New York (page 13/14), 1937] The theory of time scales, which has recently received a lot of attention, was introduced by Stefan Hilger in his PhD thesis [159] in 1988 (supervised by Bernd Aulbach) in order to unify continuous and discrete analysis. This book is an intro duction to the study of dynamic equations on time scales. Many results concerning differential equations carryover quite easily to corresponding results for difference equations, while other results seem to be completely different in nature from their continuous counterparts. The study of dynamic equations on time scales reveals such discrepancies, and helps avoid proving results twice, once for differential equa tions and once for difference equations. The general idea is to prove a result for a dynamic equation where the domain of the unknown function is a so-called time scale, which is an arbitrary nonempty closed subset of the reals.

Functional Dynamic Equations on Time Scales

Svetlin G. Georgiev

Author : Svetlin G. Georgiev
Publisher : Springer
Page : 885 pages
File Size : 50,7 Mb
Release : 2019-05-03
Category : Mathematics
ISBN : 9783030154202

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Functional Dynamic Equations on Time Scales by Svetlin G. Georgiev Pdf

This book is devoted to the qualitative theory of functional dynamic equations on time scales, providing an overview of recent developments in the field as well as a foundation to time scales, dynamic systems, and functional dynamic equations. It discusses functional dynamic equations in relation to mathematical physics applications and problems, providing useful tools for investigation for oscillations and nonoscillations of the solutions of functional dynamic equations on time scales. Practice problems are presented throughout the book for use as a graduate-level textbook and as a reference book for specialists of several disciplines, such as mathematics, physics, engineering, and biology.

Stability, Periodicity and Boundedness in Functional Dynamical Systems on Time Scales

Murat Adıvar,Youssef N. Raffoul

Author : Murat Adıvar,Youssef N. Raffoul
Publisher : Springer Nature
Page : 416 pages
File Size : 44,6 Mb
Release : 2020-04-23
Category : Mathematics
ISBN : 9783030421175

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Stability, Periodicity and Boundedness in Functional Dynamical Systems on Time Scales by Murat Adıvar,Youssef N. Raffoul Pdf

Motivated by recent increased activity of research on time scales, the book provides a systematic approach to the study of the qualitative theory of boundedness, periodicity and stability of Volterra integro-dynamic equations on time scales. Researchers and graduate students who are interested in the method of Lyapunov functions/functionals, in the study of boundedness of solutions, in the stability of the zero solution, or in the existence of periodic solutions should be able to use this book as a primary reference and as a resource of latest findings. This book contains many open problems and should be of great benefit to those who are pursuing research in dynamical systems or in Volterra integro-dynamic equations on time scales with or without delays. Great efforts were made to present rigorous and detailed proofs of theorems. The book should serve as an encyclopedia on the construction of Lyapunov functionals in analyzing solutions of dynamical systems on time scales. The book is suitable for a graduate course in the format of graduate seminars or as special topics course on dynamical systems. The book should be of interest to investigators in biology, chemistry, economics, engineering, mathematics and physics.

Conformable Dynamic Equations on Time Scales

Douglas R. Anderson,Svetlin G. Georgiev

Author : Douglas R. Anderson,Svetlin G. Georgiev
Publisher : CRC Press
Page : 347 pages
File Size : 47,5 Mb
Release : 2020-08-29
Category : Mathematics
ISBN : 9781000093933

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Conformable Dynamic Equations on Time Scales by Douglas R. Anderson,Svetlin G. Georgiev Pdf

The concept of derivatives of non-integer order, known as fractional derivatives, first appeared in the letter between L’Hopital and Leibniz in which the question of a half-order derivative was posed. Since then, many formulations of fractional derivatives have appeared. Recently, a new definition of fractional derivative, called the "fractional conformable derivative," has been introduced. This new fractional derivative is compatible with the classical derivative and it has attracted attention in areas as diverse as mechanics, electronics, and anomalous diffusion. Conformable Dynamic Equations on Time Scales is devoted to the qualitative theory of conformable dynamic equations on time scales. This book summarizes the most recent contributions in this area, and vastly expands on them to conceive of a comprehensive theory developed exclusively for this book. Except for a few sections in Chapter 1, the results here are presented for the first time. As a result, the book is intended for researchers who work on dynamic calculus on time scales and its applications. Features Can be used as a textbook at the graduate level as well as a reference book for several disciplines Suitable for an audience of specialists such as mathematicians, physicists, engineers, and biologists Contains a new definition of fractional derivative About the Authors Douglas R. Anderson is professor and chair of the mathematics department at Concordia College, Moorhead. His research areas of interest include dynamic equations on time scales and Ulam-type stability of difference and dynamic equations. He is also active in investigating the existence of solutions for boundary value problems. Svetlin G. Georgiev is currently professor at Sorbonne University, Paris, France and works in various areas of mathematics. He currently focuses on harmonic analysis, partial differential equations, ordinary differential equations, Clifford and quaternion analysis, dynamic calculus on time scales, and integral equations.

Conformable Dynamic Equations on Time Scales

Douglas R. Anderson,Svetlin G. Georgiev

Author : Douglas R. Anderson,Svetlin G. Georgiev
Publisher : CRC Press
Page : 131 pages
File Size : 47,5 Mb
Release : 2020-08-29
Category : Mathematics
ISBN : 9781000094114

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Conformable Dynamic Equations on Time Scales by Douglas R. Anderson,Svetlin G. Georgiev Pdf

The concept of derivatives of non-integer order, known as fractional derivatives, first appeared in the letter between L’Hopital and Leibniz in which the question of a half-order derivative was posed. Since then, many formulations of fractional derivatives have appeared. Recently, a new definition of fractional derivative, called the "fractional conformable derivative," has been introduced. This new fractional derivative is compatible with the classical derivative and it has attracted attention in areas as diverse as mechanics, electronics, and anomalous diffusion. Conformable Dynamic Equations on Time Scales is devoted to the qualitative theory of conformable dynamic equations on time scales. This book summarizes the most recent contributions in this area, and vastly expands on them to conceive of a comprehensive theory developed exclusively for this book. Except for a few sections in Chapter 1, the results here are presented for the first time. As a result, the book is intended for researchers who work on dynamic calculus on time scales and its applications. Features Can be used as a textbook at the graduate level as well as a reference book for several disciplines Suitable for an audience of specialists such as mathematicians, physicists, engineers, and biologists Contains a new definition of fractional derivative About the Authors Douglas R. Anderson is professor and chair of the mathematics department at Concordia College, Moorhead. His research areas of interest include dynamic equations on time scales and Ulam-type stability of difference and dynamic equations. He is also active in investigating the existence of solutions for boundary value problems. Svetlin G. Georgiev is currently professor at Sorbonne University, Paris, France and works in various areas of mathematics. He currently focuses on harmonic analysis, partial differential equations, ordinary differential equations, Clifford and quaternion analysis, dynamic calculus on time scales, and integral equations.

Dynamic Inequalities On Time Scales

Ravi Agarwal,Donal O'Regan,Samir Saker

Author : Ravi Agarwal,Donal O'Regan,Samir Saker
Publisher : Springer
Page : 264 pages
File Size : 52,5 Mb
Release : 2014-10-30
Category : Mathematics
ISBN : 9783319110028

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Dynamic Inequalities On Time Scales by Ravi Agarwal,Donal O'Regan,Samir Saker Pdf

This is a monograph devoted to recent research and results on dynamic inequalities on time scales. The study of dynamic inequalities on time scales has been covered extensively in the literature in recent years and has now become a major sub-field in pure and applied mathematics. In particular, this book will cover recent results on integral inequalities, including Young's inequality, Jensen's inequality, Holder's inequality, Minkowski's inequality, Steffensen's inequality, Hermite-Hadamard inequality and Čebyšv's inequality. Opial type inequalities on time scales and their extensions with weighted functions, Lyapunov type inequalities, Halanay type inequalities for dynamic equations on time scales, and Wirtinger type inequalities on time scales and their extensions will also be discussed here in detail.

Advances in Dynamic Equations on Time Scales

Martin Bohner,Allan C. Peterson

Author : Martin Bohner,Allan C. Peterson
Publisher : Birkhauser
Page : 366 pages
File Size : 52,6 Mb
Release : 2003
Category : Mathematics
ISBN : UOM:39015056177119

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Advances in Dynamic Equations on Time Scales by Martin Bohner,Allan C. Peterson Pdf

Excellent introductory material on the calculus of time scales and dynamic equations.; Numerous examples and exercises illustrate the diverse application of dynamic equations on time scales.; Unified and systematic exposition of the topics allows good transitions from chapter to chapter.; Contributors include Anderson, M. Bohner, Davis, Dosly, Eloe, Erbe, Guseinov, Henderson, Hilger, Hilscher, Kaymakcalan, Lakshmikantham, Mathsen, and A. Peterson, founders and leaders of this field of study.; Useful as a comprehensive resource of time scales and dynamic equations for pure and applied mathematicians.; Comprehensive bibliography and index complete this text.

Dynamic Equations on Time Scales and Applications

Ravi P Agarwal,Bipan Hazarika,Sanket Tikare

Author : Ravi P Agarwal,Bipan Hazarika,Sanket Tikare
Publisher : CRC Press
Page : 435 pages
File Size : 45,5 Mb
Release : 2024-10-18
Category : Mathematics
ISBN : 9781040103739

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Dynamic Equations on Time Scales and Applications by Ravi P Agarwal,Bipan Hazarika,Sanket Tikare Pdf

This book presents the theory of dynamic equations on time scales and applications, providing an overview of recent developments in the foundations of the field as well as its applications. It discusses the recent results related to the qualitative properties of solutions like existence and uniqueness, stability, continuous dependence, controllability, oscillations, etc. Presents cutting-edge research trends of dynamic equations and recent advances in contemporary research on the topic of time scales Connects several new areas of dynamic equations on time scales with applications in different fields Includes mathematical explanation from the perspective of existing knowledge of dynamic equations on time scales Offers several new recently developed results, which are useful for the mathematical modeling of various phenomena Useful for several interdisciplinary fields like economics, biology, and population dynamics from the perspective of new trends The text is for postgraduate students, professionals, and academic researchers working in the fields of Applied Mathematics

Uncertain Dynamical Systems

A.A. Martynyuk,Yu. A. Martynyuk-Chernienko

Author : A.A. Martynyuk,Yu. A. Martynyuk-Chernienko
Publisher : CRC Press
Page : 310 pages
File Size : 43,9 Mb
Release : 2011-11-28
Category : Mathematics
ISBN : 9781439876879

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Uncertain Dynamical Systems by A.A. Martynyuk,Yu. A. Martynyuk-Chernienko Pdf

This self-contained book provides systematic instructive analysis of uncertain systems of the following types: ordinary differential equations, impulsive equations, equations on time scales, singularly perturbed differential equations, and set differential equations. Each chapter contains new conditions of stability of unperturbed motion of the abo

Complex-valued Neural Networks

Akira Hirose

Author : Akira Hirose
Publisher : World Scientific
Page : 387 pages
File Size : 43,5 Mb
Release : 2003
Category : Computers
ISBN : 9789812384645

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Complex-valued Neural Networks by Akira Hirose Pdf

In recent years, complex-valued neural networks have widened the scope of application in optoelectronics, imaging, remote sensing, quantum neural devices and systems, spatiotemporal analysis of physiological neural systems, and artificial neural information processing. In this first-ever book on complex-valued neural networks, the most active scientists at the forefront of the field describe theories and applications from various points of view to provide academic and industrial researchers with a comprehensive understanding of the fundamentals, features and prospects of the powerful complex-valued networks.

Stability of Dynamical Systems

Xiaoxin Liao,L.Q. Wang,P. Yu

Author : Xiaoxin Liao,L.Q. Wang,P. Yu
Publisher : Elsevier
Page : 719 pages
File Size : 40,7 Mb
Release : 2007-08-01
Category : Mathematics
ISBN : 9780080550619

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Stability of Dynamical Systems by Xiaoxin Liao,L.Q. Wang,P. Yu Pdf

The main purpose of developing stability theory is to examine dynamic responses of a system to disturbances as the time approaches infinity. It has been and still is the object of intense investigations due to its intrinsic interest and its relevance to all practical systems in engineering, finance, natural science and social science. This monograph provides some state-of-the-art expositions of major advances in fundamental stability theories and methods for dynamic systems of ODE and DDE types and in limit cycle, normal form and Hopf bifurcation control of nonlinear dynamic systems. Presents comprehensive theory and methodology of stability analysis Can be used as textbook for graduate students in applied mathematics, mechanics, control theory, theoretical physics, mathematical biology, information theory, scientific computation Serves as a comprehensive handbook of stability theory for practicing aerospace, control, mechanical, structural, naval and civil engineers

Dynamic Systems on Measure Chains

V. Lakshmikantham,S. Sivasundaram,B. Kaymakcalan

Author : V. Lakshmikantham,S. Sivasundaram,B. Kaymakcalan
Publisher : Springer Science & Business Media
Page : 300 pages
File Size : 51,6 Mb
Release : 2013-06-29
Category : Mathematics
ISBN : 9781475724493

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Dynamic Systems on Measure Chains by V. Lakshmikantham,S. Sivasundaram,B. Kaymakcalan Pdf

From a modelling point of view, it is more realistic to model a phenomenon by a dynamic system which incorporates both continuous and discrete times, namely, time as an arbitrary closed set of reals called time-scale or measure chain. It is therefore natural to ask whether it is possible to provide a framework which permits us to handle both dynamic systems simultaneously so that one can get some insight and a better understanding of the subtle differences of these two different systems. The answer is affirmative, and recently developed theory of dynamic systems on time scales offers the desired unified approach. In this monograph, we present the current state of development of the theory of dynamic systems on time scales from a qualitative point of view. It consists of four chapters. Chapter one develops systematically the necessary calculus of functions on time scales. In chapter two, we introduce dynamic systems on time scales and prove the basic properties of solutions of such dynamic systems. The theory of Lyapunov stability is discussed in chapter three in an appropriate setup. Chapter four is devoted to describing several different areas of investigations of dynamic systems on time scales which will provide an exciting prospect and impetus for further advances in this important area which is very new. Some important features of the monograph are as follows: It is the first book that is dedicated to a systematic development of the theory of dynamic systems on time scales which is of recent origin. It demonstrates the interplay of the two different theories, namely, the theory of continuous and discrete dynamic systems, when imbedded in one unified framework. It provides an impetus to investigate in the setup of time scales other important problems which might offer a better understanding of the intricacies of a unified study.£/LIST£ Audience: The readership of this book consists of applied mathematicians, engineering scientists, research workers in dynamic systems, chaotic theory and neural nets.

Combined Measure and Shift Invariance Theory of Time Scales and Applications

Chao Wang,Ravi P. Agarwal

Author : Chao Wang,Ravi P. Agarwal
Publisher : Springer Nature
Page : 443 pages
File Size : 42,9 Mb
Release : 2022-09-22
Category : Mathematics
ISBN : 9783031116193

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Combined Measure and Shift Invariance Theory of Time Scales and Applications by Chao Wang,Ravi P. Agarwal Pdf

This monograph is devoted to developing a theory of combined measure and shift invariance of time scales with the related applications to shift functions and dynamic equations. The study of shift closeness of time scales is significant to investigate the shift functions such as the periodic functions, the almost periodic functions, the almost automorphic functions, and their generalizations with many relevant applications in dynamic equations on arbitrary time scales. First proposed by S. Hilger, the time scale theory—a unified view of continuous and discrete analysis—has been widely used to study various classes of dynamic equations and models in real-world applications. Measure theory based on time scales, in its turn, is of great power in analyzing functions on time scales or hybrid domains. As a new and exciting type of mathematics—and more comprehensive and versatile than the traditional theories of differential and difference equations—, the time scale theory can precisely depict the continuous-discrete hybrid processes and is an optimal way forward for accurate mathematical modeling in applied sciences such as physics, chemical technology, population dynamics, biotechnology, and economics and social sciences. Graduate students and researchers specializing in general dynamic equations on time scales can benefit from this work, fostering interest and further research in the field. It can also serve as reference material for undergraduates interested in dynamic equations on time scales. Prerequisites include familiarity with functional analysis, measure theory, and ordinary differential equations.

Dynamic Equations on Time Scales and Applications

Ravi P Agarwal,Bipan Hazarika,Sanket Tikare

Author : Ravi P Agarwal,Bipan Hazarika,Sanket Tikare
Publisher : CRC Press
Page : 599 pages
File Size : 43,7 Mb
Release : 2024-10-18
Category : Mathematics
ISBN : 9781040103753

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Dynamic Equations on Time Scales and Applications by Ravi P Agarwal,Bipan Hazarika,Sanket Tikare Pdf

This book presents the theory of dynamic equations on time scales and applications, providing an overview of recent developments in the foundations of the field as well as its applications. It discusses the recent results related to the qualitative properties of solutions like existence and uniqueness, stability, continuous dependence, controllability, oscillations, etc. • Presents cutting-edge research trends of dynamic equations and recent advances in contemporary research on the topic of time scales • Connects several new areas of dynamic equations on time scales with applications in different fields • Includes mathematical explanation from the perspective of existing knowledge of dynamic equations on time scales • Offers several new recently developed results, which are useful for the mathematical modeling of various phenomena • Useful for several interdisciplinary fields like economics, biology, and population dynamics from the perspective of new trends The text is for postgraduate students, professionals, and academic researchers working in the fields of Applied Mathematics

Stability Theory For Dynamic Equations On Time Scales

Stability Theory for Dynamic Equations on Time Scales by Anatoly A. Martynyuk Pdf

Dynamic Equations on Time Scales by Martin Bohner,Allan Peterson Pdf

Functional Dynamic Equations on Time Scales by Svetlin G. Georgiev Pdf

Stability, Periodicity and Boundedness in Functional Dynamical Systems on Time Scales by Murat Adıvar,Youssef N. Raffoul Pdf

Conformable Dynamic Equations on Time Scales by Douglas R. Anderson,Svetlin G. Georgiev Pdf

Conformable Dynamic Equations on Time Scales by Douglas R. Anderson,Svetlin G. Georgiev Pdf

Dynamic Inequalities On Time Scales by Ravi Agarwal,Donal O'Regan,Samir Saker Pdf

Advances in Dynamic Equations on Time Scales by Martin Bohner,Allan C. Peterson Pdf

Dynamic Equations on Time Scales and Applications by Ravi P Agarwal,Bipan Hazarika,Sanket Tikare Pdf

Uncertain Dynamical Systems by A.A. Martynyuk,Yu. A. Martynyuk-Chernienko Pdf

Complex-valued Neural Networks by Akira Hirose Pdf

Stability of Dynamical Systems by Xiaoxin Liao,L.Q. Wang,P. Yu Pdf

Dynamic Systems on Measure Chains by V. Lakshmikantham,S. Sivasundaram,B. Kaymakcalan Pdf

Combined Measure and Shift Invariance Theory of Time Scales and Applications by Chao Wang,Ravi P. Agarwal Pdf

Dynamic Equations on Time Scales and Applications by Ravi P Agarwal,Bipan Hazarika,Sanket Tikare Pdf

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