Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures

Advances in discrete mathematics and applications

Book 5
CRC Press
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Extending and generalizing the results of rational equations,

Dynamics of Third Order Rational Difference Equations with Open Problems and Conjectures focuses on the boundedness nature of solutions, the global stability of equilibrium points, the periodic character of solutions, and the convergence to periodic solutions, including their periodic trichotomies. The book also provides numerous thought-provoking open problems and conjectures on the boundedness character, global stability, and periodic behavior of solutions of rational difference equations.

After introducing several basic definitions and general results, the authors examine 135 special cases of rational difference equations that have only bounded solutions and the equations that have unbounded solutions in some range of their parameters. They then explore the seven known nonlinear periodic trichotomies of third order rational difference equations. The main part of the book presents the known results of each of the 225 special cases of third order rational difference equations. In addition, the appendices supply tables that feature important information on these cases as well as on the boundedness character of all fourth order rational difference equations.

A Framework for Future Research

The theory and techniques developed in this book to understand the dynamics of rational difference equations will be useful in analyzing the equations in any mathematical model that involves difference equations. Moreover, the stimulating conjectures will promote future investigations in this fascinating, yet surprisingly little known area of research.

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Additional Information

Publisher
CRC Press
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Published on
Nov 16, 2007
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Pages
576
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ISBN
9781584887669
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Best For
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Language
English
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Genres
Mathematics / Applied
Mathematics / Differential Equations / General
Science / Physics / Mathematical & Computational
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Content Protection
This content is DRM protected.
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Eligible for Family Library

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Book 4
Sharkovsky's Theorem, Li and Yorke's "period three implies chaos" result, and the (3x+1) conjecture are beautiful and deep results that demonstrate the rich periodic character of first-order, nonlinear difference equations. To date, however, we still know surprisingly little about higher-order nonlinear difference equations.

During the last ten years, the authors of this book have been fascinated with discovering periodicities in equations of higher order which for certain values of their parameters have one of the following characteristics:

1. Every solution of the equation is periodic with the same period.
2. Every solution of the equation is eventually periodic with a prescribed period.
3. Every solution of the equation converges to a periodic solution with the same period.

This monograph presents their findings along with some thought-provoking questions and many open problems and conjectures worthy of investigation. The authors also propose investigation of the global character of solutions of these equations for other values of their parameters and working toward a more complete picture of the global behavior of their solutions.

With the results and discussions it presents, Periodicities in Nonlinear Difference Equations places a few more stones in the foundation of the basic theory of nonlinear difference equations. Researchers and graduate students working in difference equations and discrete dynamical systems will find much to intrigue them and inspire further work in this area.
Book 4
Sharkovsky's Theorem, Li and Yorke's "period three implies chaos" result, and the (3x+1) conjecture are beautiful and deep results that demonstrate the rich periodic character of first-order, nonlinear difference equations. To date, however, we still know surprisingly little about higher-order nonlinear difference equations.

During the last ten years, the authors of this book have been fascinated with discovering periodicities in equations of higher order which for certain values of their parameters have one of the following characteristics:

1. Every solution of the equation is periodic with the same period.
2. Every solution of the equation is eventually periodic with a prescribed period.
3. Every solution of the equation converges to a periodic solution with the same period.

This monograph presents their findings along with some thought-provoking questions and many open problems and conjectures worthy of investigation. The authors also propose investigation of the global character of solutions of these equations for other values of their parameters and working toward a more complete picture of the global behavior of their solutions.

With the results and discussions it presents, Periodicities in Nonlinear Difference Equations places a few more stones in the foundation of the basic theory of nonlinear difference equations. Researchers and graduate students working in difference equations and discrete dynamical systems will find much to intrigue them and inspire further work in this area.
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