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Difference Equations and Their Applications

  • Book
  • © 1993

Overview

Authors:
  1. A. N. Sharkovsky
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  2. Yu. L. Maistrenko
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  3. E. Yu. Romanenko

    1. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev, Ukraine

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Part of the book series: Mathematics and Its Applications (MAIA, volume 250)

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Table of contents (13 chapters)

  1. Front Matter

    Pages i-xii
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  2. Introduction

    1. Introduction

      • A. N. Sharkovsky, Yu. L. Maistrenko, E. Yu. Romanenko
      Pages 1-13

  3. One-Dimensional Dynamical Systems

    1. Introduction to the Theory of Dynamical Systems

      • A. N. Sharkovsky, Yu. L. Maistrenko, E. Yu. Romanenko
      Pages 15-43

    2. Periodic Trajectories

      • A. N. Sharkovsky, Yu. L. Maistrenko, E. Yu. Romanenko
      Pages 45-70

    3. Behavior of Trajectories

      • A. N. Sharkovsky, Yu. L. Maistrenko, E. Yu. Romanenko
      Pages 71-94

    4. Dynamical Systems for U-Maps

      • A. N. Sharkovsky, Yu. L. Maistrenko, E. Yu. Romanenko
      Pages 95-124

  4. Difference Equations with Continuous Time

    1. Nonlinear Difference Equations

      • A. N. Sharkovsky, Yu. L. Maistrenko, E. Yu. Romanenko
      Pages 125-158

    2. Difference Equations with U-Nonlinearity

      • A. N. Sharkovsky, Yu. L. Maistrenko, E. Yu. Romanenko
      Pages 159-185

  5. Differential-Difference Equations

    1. Front Matter

      Pages 187-187
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    2. Completely Integrable Differential-Difference Equations

      • A. N. Sharkovsky, Yu. L. Maistrenko, E. Yu. Romanenko
      Pages 188-222

    3. Differential-Difference Equations Close to Difference Ones

      • A. N. Sharkovsky, Yu. L. Maistrenko, E. Yu. Romanenko
      Pages 223-237

    4. Singularly Perturbed Differential-Difference Equations

      • A. N. Sharkovsky, Yu. L. Maistrenko, E. Yu. Romanenko
      Pages 239-272

  6. Boundary-Value Problems for Hyperbolic Systems of Partial Differential Equations

    1. Front Matter

      Pages 273-273
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    2. Reduction of Boundary Value Problems to Difference and Differential-Difference Equations

      • A. N. Sharkovsky, Yu. L. Maistrenko, E. Yu. Romanenko
      Pages 274-284

    3. Boundary-Value Problem for a System with Small Parameter

      • A. N. Sharkovsky, Yu. L. Maistrenko, E. Yu. Romanenko
      Pages 285-304

    4. Boundary-Value Problem for Systems with Two Spatial Variables

      • A. N. Sharkovsky, Yu. L. Maistrenko, E. Yu. Romanenko
      Pages 305-334

  7. Back Matter

    Pages 335-358
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Keywords

About this book

The theory of difference equations is now enjoying a period of Renaissance. Witness the large number of papers in which problems, having at first sight no common features, are reduced to the investigation of subsequent iterations of the maps f· IR. m ~ IR. m, m > 0, or (which is, in fact, the same) to difference equations The world of difference equations, which has been almost hidden up to now, begins to open in all its richness. Those experts, who usually use differential equations and, in fact, believe in their universality, are now discovering a completely new approach which re­ sembles the theory of ordinary differential equations only slightly. Difference equations, which reflect one of the essential properties of the real world-its discreteness-rightful­ ly occupy a worthy place in mathematics and its applications. The aim of the present book is to acquaint the reader with some recently discovered and (at first sight) unusual properties of solutions for nonlinear difference equations. These properties enable us to use difference equations in order to model complicated os­ cillating processes (this can often be done in those cases when it is difficult to apply ordinary differential equations). Difference equations are also a useful tool of syn­ ergetics- an emerging science concerned with the study of ordered structures. The application of these equations opens up new approaches in solving one of the central problems of modern science-the problem of turbulence.

Reviews

' I enjoyed reading this book, and i am happy to recommend it to all those interested in finding out more about this fascinating but rather unfamiliar branch of dynamical systems.' Bulletin London Math. Soc. 27 1995

Authors and Affiliations

  • Institute of Mathematics, Ukrainian Academy of Sciences, Kiev, Ukraine

    E. Yu. Romanenko

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