Form Symmetries and Reduction of Order in Difference Equations

Form Symmetries and Reduction of Order in Difference Equations presents a new approach to the formulation and analysis of difference equations in which the underlying space is typically an algebraic group. In some problems and applications, an additional algebraic or topological structure is assumed in order to define equations and obtain significant results about them. Reflecting the author’s past research experience, the majority of examples involve equations in finite dimensional Euclidean spaces.

The book first introduces difference equations on groups, building a foundation for later chapters and illustrating the wide variety of possible formulations and interpretations of difference equations that occur in concrete contexts. The author then proposes a systematic method of decomposition for recursive difference equations that uses a semiconjugate relation between maps. Focusing on large classes of difference equations, he shows how to find the semiconjugate relations and accompanying factorizations of two difference equations with strictly lower orders. The final chapter goes beyond semiconjugacy by extending the fundamental ideas based on form symmetries to nonrecursive difference equations.

With numerous examples and exercises, this book is an ideal introduction to an exciting new domain in the area of difference equations. It takes a fresh and all-inclusive look at difference equations and develops a systematic procedure for examining how these equations are constructed and solved.

Hassan Sedaghat is a professor of mathematics at Virginia Commonwealth University. His research interests include difference equations and discrete dynamical systems and their applications in mathematics, economics, biology, and medicine.

Introduction

Difference Equations on Groups Basic definitions One equation, many interpretations Examples of difference equations on groups

Semiconjugate Factorization and Reduction of Order Semiconjugacy and ordering of mapsForm symmetries and SC factorizationsOrder-reduction typesSC factorizations as triangular systems Order-preserving form symmetries

Homogeneous Equations of Degree One Homogeneous equations on groups Characteristic form symmetry of HD1 equations Reductions of order in HD1 equations Absolute value equation

Type-(k,1) Reductions Invertible-map criterion Identity form symmetry Inversion form symmetry Discrete Riccati equation of order two Linear form symmetryDifference equations with linear argumentsField-inverse form symmetry

Type-(1,k) Reductions Linear form symmetry revisited Separable difference equationsEquations with exponential and power functions

Time-Dependent Form SymmetriesThe semiconjugate relation and factorization Invertible-map criterion revisited Time-dependent linear form symmetry SC factorization of linear equations

Nonrecursive Difference EquationsExamples and discussion Form symmetries, factors, and cofactors Semi-invertible map criterion Quadratic difference equationsAn order-preserving form symmetry

Appendix: Asymptotic Stability on the Real Line

References
Index

Notes and Problems appear at the end of each chapter.